Sum of Infinite Geometric Series

Previously we learned about the technique about the sum of geometric series, this time let’s tackle the sum of infinite geometric series. I have a secret to confess, series is my favorites topic in algebra especially telescoping series. I first heard about that series when I was 21. The time I joined a very active math group in facebook called Elite Math Circle. One of the applications of this series is to find the exact value of repeating decimals. I will also give example about that here. Let’s start!

Formula:

Given an infinite geometric sum expression a+ar+ar^2+\dots where 0<r<1, the sum(Sn) can be expressed as follows:

S_n=a+ar+ar^2+ \dots

S_n=\displaystyle\frac{a}{1-r}

 

Sample Problem 1:

Find the sum of 2+1+\frac{1}{2}+\frac{1}{4}+\dots

Solution:

Using the formula,

S_n=\displaystyle\frac{a}{1-r}

where r=1/2 and a=2

S_{\infty}=\displaystyle\frac{2}{1-\frac{1}{2}}

S_{\infty}=4

 

Sample Problem 2:

Ball rolled 16 cm in the first second, 12 cm in the second seconds, 9 cm on the third. Find the total distance travelled by the ball until it stops. Assume that the motion of the ball continues until it stops.

Solution:

This is another application of sum of infinite geometric series. If you think like a physicist, you might be thinking how much is the deceleration of the ball or the reason why the ball is decelerating. But set that aside for a while.

Using our formula,

S_n=\displaystyle\frac{a}{1-r}

where r=9/12=12/16=3/4, and a=16

S_{\infty}=\displaystyle\frac{16}{1-\frac{3}{4}}

S_{\infty}=64 cm

Sample Problem 3:

Find the exact value of 0.1212121212….

Solution:

There is an easy way to solve this problem. The process was discussed in detailed here.

But using the sum of geometric series, here is the process.

We can express the following in this form.

0.12+0.0012+0.000012+\dots where r=0.01

Using the formula,

S_n=\displaystyle\frac{a}{1-r}

S_n=\displaystyle\frac{0.12}{1-0.01}

S_n=\displaystyle\frac{0.12}{.99}=\displaystyle\frac{0.12}{.99}=\displaystyle\frac{4}{33}

 

Sample Problem 4:

Find the exact value of 0.345454545…

Solution:

Rewrite the given as the sum of 0.3+0.045+0.00045+0.0000045+\dots

From the term 0.045 onwards the series forms a geometric progression with r=0.01

Using the formula,

S_n=\displaystyle\frac{a}{1-r}

where a=0.045 and r=0.01

S_n=0.3+\displaystyle\frac{0.045}{1-0.01}

S_n=\displaystyle\frac{19}{55}

 

Sample Problem 5:

Find the sum of the series

1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{12}+\dots

Solution:

Rearrange the series in this manner,

(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots)+(\frac{1}{3}+\frac{1}{6}+\frac{1}{12}+\dots)

These are two infinite geometric series so we can use our formula.

S_n=\displaystyle\frac{a}{1-r}+\displaystyle\frac{a}{1-r}

S_n=\displaystyle\frac{1}{1-\frac{1}{2}}+\displaystyle\frac{\frac{1}{3}}{1-\frac{1}{2}}

S_n=\displaystyle\frac{8}{3}

 

Sample Problem 6:

A rubber ball is made to fall from a height of 50 feet and is observed to rebound 2/3 of the distance it falls. How far will the ball travel before coming to rest if the ball continues to fall in this manner?

Solution:

Sum of geometric Series ball

The movement of the ball must be vertical. Just think that the figure above is a projected view of the movement of the ball. We’ll be having a hard time to show the movement if the lines are overlapping. The red diagonal must be vertical so I projected another line on the left.

The total distance travelled by the ball must be the upward and downward path travelled by the ball.

Let Sd be total distance travelled by the ball downward and Su is the total distance travelled by the ball upward.

S_n=\displaystyle\frac{a}{1-r}

S_d=\displaystyle\frac{50}{1-\frac{2}{3}}

S_d=150 ft

 

S_u=\displaystyle\frac{a}{1-r} ,

this time our a=50(2/3)=100/3.

S_u=\displaystyle\frac{\frac{100}{3}}{1-\frac{2}{3}}

S_u=100 ft

Total distance = Sd+Su = 150+100=250 ft.

Dan

Dan

Blogger and a Math enthusiast. Has no interest in Mathematics until MMC came. Aside from doing math, he also loves to travel and watch movies.
Dan

Latest posts by Dan (see all)

You may also like...

116 Responses

  1. Shani82 says:

    I see you don’t monetize your blog. You can make
    extra income, your niche is good for one method, just search in google – reselling online by Riggso

  2. Apolonia says:

    What i don’t understood is actually how you’re not actually much more well-liked than you might be right now. You are so intelligent. You realize therefore significantly relating to this subject, made me personally consider it from a lot of varied angles. Its like women and men aren’t fascinated unless it’s one thing to do with Lady gaga! Your own stuffs nice. Always maintain it up!

  3. I’m impressed, I need to say. Actually hardly ever do I encounter a blog that’s both educative and entertaining, and let me tell you, you have got hit the nail on the head. Your idea is outstanding; the concern is something that not sufficient folks are speaking intelligently about. I’m incredibly pleased that I stumbled across this in my search for something relating to this.

  4. lucy ann says:

    b8t5Bc I truly appreciate this article.Much thanks again. Much obliged.

  5. NvYL5p Really informative article post.Thanks Again. Really Great.

  6. Many thanks for sharing this very good piece. Very inspiring! (as always, btw)

  7. Im obliged for the post.Really thank you!

  8. I really liked your article.Really looking forward to read more. Awesome.

  9. bed pillows says:

    Really enjoyed this article post. Keep writing.

  10. I think other web site proprietors should take this website as an model, very clean and magnificent user friendly style and design, as well as the content. You are an expert in this topic!

  11. This is one awesome blog.Thanks Again.

  12. Very good post.Really looking forward to read more.

  13. Rattling clean internet internet site , appreciate it for this post.

  14. of course, research is paying off. I enjoy you sharing your point of view.. Great thoughts you have here.. I value you discussing your point of view..

  15. I value the blog post.Thanks Again. Will read on

  16. Your method of telling everything in this article is genuinely pleasant, all can without difficulty know it, Thanks a lot.

  17. candle says:

    usually posts some incredibly fascinating stuff like this. If you

  18. UoN says:

    It as exhausting to search out educated folks on this subject, however you sound like you recognize what you are speaking about! Thanks

  19. bbw wife says:

    If you are going for best contents like me, only pay a quick visit this website every day as it offers quality contents, thanks

  20. Geschichte says:

    You made some good points there. I looked on the internet for the topic and found most people will agree with your site.

  21. My brother recommended I might like this website. He was totally right. This post truly made my day. You can not imagine simply how much time I had spent for this info! Thanks!

  22. when opening in Internet Explorer, it has some overlapping. I just wanted to give you a quick heads up! Other then that,

  23. Very neat article post.Thanks Again. Really Great.

  24. nikon d7200 says:

    Well I really liked reading it. This article provided by you is very useful for accurate planning.

  25. You are my breathing in, I possess few web logs and rarely run out from to brand.

  26. A round of applause for your blog.Much thanks again. Really Great.

  27. amiclubwear says:

    I truly appreciate this blog post. Really Cool.

  28. You made some respectable factors there. I seemed on the web for the difficulty and located most people will go together with together with your website.

  29. archangel says:

    Pretty! This has been an extremely wonderful post. Thanks for supplying this info.

  30. This is one awesome blog post.

  31. luggage says:

    Only a smiling visitant here to share the love (:, btw outstanding style and design. Reading well is one of the great pleasures that solitude can afford you. by Harold Bloom.

  32. pretty beneficial stuff, overall I think this is worthy of a bookmark, thanks

  33. Im thankful for the article post.Much thanks again. Really Cool.

  34. Way cool! Some very valid points! I appreciate you penning this post plus the rest of the site is also really good.

  35. Pork says:

    Thanks for sharing, this is a fantastic blog.Much thanks again. Much obliged.

  36. Photography says:

    Great article.Thanks Again. Really Great.

  37. Friv says:

    Very interesting points you have noted, thanks for putting up.

  38. teachers says:

    I appreciate you sharing this blog post.Really thank you! Great.

  39. Thank you for your article.Really thank you! Fantastic.

  40. Pretty! This has been a really wonderful post. Thanks for providing this info.

  41. Thank you for your blog post.Much thanks again. Want more.

  42. Regards for this wonderful post, I am glad I discovered this web site on yahoo.

  43. Fashion Show says:

    It as nearly impossible to find experienced people for this topic, but you sound like you know what you are talking about! Thanks

  44. It as not that I want to duplicate your web-site, but I really like the style. Could you let me know which style are you using? Or was it custom made?

  45. Howdy! Would you mind if I share your blog with my facebook
    group? There’s a lot of folks that I think would really enjoy your content.

    Please let me know. Cheers

  46. Wow! This could be one particular of the most helpful blogs We ave ever arrive across on this subject. Actually Great. I am also a specialist in this topic so I can understand your hard work.

  47. Some really fantastic content on this website , thanks for contribution.

  48. Im obliged for the article. Will read on…

  49. I will right away snatch your rss as I can not in finding your email subscription link or newsletter service. Do you have any? Please let me recognize in order that I may just subscribe. Thanks.

  50. Looking forward to reading more. Great article post.Thanks Again. Keep writing.

  51. Thanks again for the article.Thanks Again. Keep writing.

  52. I think this is a real great post. Really Great.

  53. I simply could not depart your site prior to suggesting that I extremely loved the usual info a person supply on your guests? Is going to be again continuously to investigate cross-check new posts

  54. Looking forward to reading more. Great post.Really thank you! Really Great.

  55. Really enjoyed this post.Really looking forward to read more. Awesome.

  56. I think this is a real great blog post. Great.

  57. Cathey says:

    This page truly has all the info I needed about this subject and didn at know who to ask.

  58. I truly appreciate this blog.Really thank you! Will read on…

  59. Elmer says:

    redirected here Where can I find the best online creative writing courses? I live in NYC so which colleges offer the best online creative writing course? If not in a college than where else?.

  60. Really appreciate you sharing this post. Keep writing.

  61. Jackeline says:

    I was able to find good information from your blog posts.

  62. Denae says:

    looked at. And on this article Referencement editorial :

  63. Im thankful for the article.Much thanks again. Really Cool.

  64. Very neat blog article. Really Cool.

  65. Music says:

    Thank you ever so for you post.Really looking forward to read more. Will read on

  66. Muchos Gracias for your blog post. Keep writing.

  67. Sick! Just received a brand-new Pearl and I can now read your blog on my phone as browser, it didn at operate on my old one.

  68. Mirella says:

    I must convey my passion for your generosity for persons who absolutely need guidance on this particular situation. Your personal dedication to getting the message along appeared to be extraordinarily practical and have in most cases made regular people like me to get to their ambitions. Your personal helpful tips and hints denotes a great deal a person like me and still more to my peers. Warm regards; from everyone of us.

  69. #leafly says:

    Thanks again for the blog post.Thanks Again. Awesome.

  70. Really informative article post. Much obliged.

  71. Im thankful for the blog post.Really thank you! Will read on

  72. This is a topic that as close to my heart Thank you! Where are your contact details though?

  73. Really informative blog post. Awesome.

  74. Just wanna comment that you have a very nice website , I enjoy the style it really stands out.

  75. Major thankies for the article post.Thanks Again. Fantastic.

  76. opslagruimte says:

    Very informative blog post.Really thank you! Will read on

  77. You made some really good points there. I checked on the internet for additional information about the issue and found most individuals will go along with your views on this website.

  78. Very neat blog.Really looking forward to read more.

  79. flexvit says:

    I truly appreciate individuals like you! Take care!!

  80. This blog is no doubt educating as well as factual. I have discovered helluva handy things out of it. I ad love to visit it again soon. Thanks a lot!

  81. Thanks for sharing, this is a fantastic blog.Thanks Again. Awesome.

  82. Muchos Gracias for your post.Really looking forward to read more. Much obliged.

  83. I truly appreciate this post.Thanks Again. Will read on

  84. Looking around I like to browse in various places on the internet, often I will go to Stumble Upon and read and check stuff out

  85. Thanks a whole lot for sharing this with all of us you really know what you are speaking about! Bookmarked. Kindly also check out my web-site =). We could possess a link exchange contract amongst us!

  86. WWW says:

    Very nice post. I just stumbled upon your blog and wanted to say that I ave truly enjoyed browsing your blog posts. In any case I all be subscribing to your feed and I hope you write again very soon!

  87. Regards for all your efforts that you have put in this. very interesting information.

  88. ebooki.net says:

    Wow! Thank you! I constantly wanted to write on my site something like that. Can I include a part of your post to my blog?

  89. Jerilyn says:

    I really liked your blog.Really looking forward to read more. Will read on

  90. 2017 says:

    Really informative blog.Much thanks again. Cool.

  91. Thanks so much for the article post. Want more.

  92. We all talk a little about what you should speak about when is shows correspondence to because Perhaps this has more than one meaning.

  93. Daoist Magic says:

    This is a really good tip especially to those new to the blogosphere. Simple but very precise info Thanks for sharing this one. A must read article!

  94. MichaAftEn says:

    Does Amoxicillin Suspension Treat Abladder Infection [url=http://cial5mg.xyz/fast-delivery-cialis.php]Fast Delivery Cialis[/url] Levitra Generique 5mg About Contact Terms Privacy Telecom Phone Sites Loans for People with Bad Credit The Digest Unsecured personal loans can provide as much as without collateral but payday loans are designated for much smaller amounts only up to. [url=http://easymoneyfast-365.com]need money now[/url] j.The Benefits include.Amoxicillin 875mg Tablet [url=http://cial5mg.xyz/cialis.php]Cialis[/url] Cialis E Viagra Online Cialis 20 Mg Walmart [url=http://cial5mg.xyz/cialis-on-line.php]Cialis On Line[/url] Acheter Cytotec En Ligne Fed Ex Inhouse Phyarmacy [url=http://xenical.ccrpdc.com/best-xenical-online.php]Best Xenical Online[/url] Cymbalta Generic Viagra Accept Paypal 215 [url=http://cial1.xyz/cialis-online-usa.php]Cialis Online Usa[/url] Cheap Viagra Kamagra From The U.K Acquistare Levitra 20mg [url=http://cial5mg.xyz/cialis-40mg.php]Cialis 40mg[/url] Vente De Viagra Cephalexin Info [url=http://cial5mg.xyz/tadalafil-tablets.php]Tadalafil Tablets[/url] Canadian Pharmacies Nexium Levitra Mas Alcohol [url=http://zol1.xyz/cheap-generic-zoloft.php]Cheap Generic Zoloft[/url] Clobetasol Overnight With Free Shipping Oldham Delta Enterprises Cialis [url=http://zol1.xyz/low-price-zoloft.php]Low Price Zoloft[/url] Propecia Dosis Achat Cialis Ligne Suisse [url=http://zol1.xyz/zoloft-free-offer.php]Zoloft Free Offer[/url] Viagra Cealis Canada Finasteride Next Day Delivery Discount [url=http://kama1.xyz/kamagra-oral-jelly.php]Kamagra Oral Jelly[/url] Cialis No Prescription From Canada Acheter Du Amoxicillin En Ligne [url=http://kama1.xyz/oral-jelly-kamagra.php]Oral Jelly Kamagra[/url] Vipps Pharmacy Selling Viagra Purchasing Levaquin Overnight [url=http://cial1.xyz/cheap-cialis-generic.php]Cheap Cialis Generic[/url] Tretinoin Clomid A Vendre [url=http://viag1.xyz/purchase-viagra-usa.php]Purchase Viagra Usa[/url] Buy Cheap Celexa Commander Baclofen 10mg [url=http://kama1.xyz/kamagra-oral-jelly.php]Kamagra Oral Jelly[/url] Todo Sobre Kamagra Canadarx Mall [url=http://viag1.xyz/cheap-viagra-generic.php]Cheap Viagra Generic[/url] Buy Now Legally Tab Doryx Free Shipping Store Xenical Orlistat 120 Mg Prix [url=http://zol1.xyz/low-price-zoloft.php]Low Price Zoloft[/url] Propecia Blisters Propecia Cesare Ragazzi [url=http://zol1.xyz/buying-zoloft-online.php]Buying Zoloft Online[/url] Precio Cialis Farmacia Comprar Viagra Cialis Levitra [url=http://accutane.ccrpdc.com/accutane-cost.php]Accutane Cost[/url] Come Ordinare Viagra Order Shipped Ups Stendra Medication Virginia Beach [url=http://viagra.ccrpdc.com/order-viagra-onlines.php]Order Viagra Onlines[/url] Achat Cytotec Espagne Propecia Veneno Venta [url=http://cial5mg.xyz/cheapest-cialis-online.php]Cheapest Cialis Online[/url] Levaquin Medication Cheap Pharmacy

  95. I really liked your blog.Really thank you! Fantastic.

  96. Really informative article.Really thank you! Awesome.

  97. this topic to be really something that I think I would never understand.

  98. I think this is a real great blog article.Really looking forward to read more. Great.

  99. Wow, this post is fastidious, my younger sister is analyzing these things, thus I am going to inform her.

  100. This is one awesome article post. Really Great.

  101. App web says:

    I think this is a real great blog post. Really Cool.

  102. E-commerce says:

    Thank you for your article post.Much thanks again. Really Great.

  103. Webdesign says:

    You should take part in a contest for one of the most useful websites on the net. I am going to highly recommend this blog!

  104. Thanks for the blog. Fantastic.

  105. You made some clear points there. I looked on the internet for the issue and found most persons will approve with your site.

  106. Thank you for your blog post.Thanks Again. Cool.

  107. Thanks again for the article.Really thank you! Fantastic.

  108. You have made some good points there. I checked on the internet for additional information about the issue and found most people will go along with your views on this site.

  109. Very neat post.Much thanks again. Awesome.

  110. Thanks again for the post. Keep writing.

  111. bioskop25 says:

    Muchos Gracias for your article post.Thanks Again. Much obliged.

  112. I value the blog post. Really Cool.

  113. You have noted very interesting points! ps decent site.

  114. Stop Protesting And Start your own special men Advertising and marketing campaign In exchange

  115. You ought to be a part of a contest for one of the best sites on the net.

Leave a Reply

Your email address will not be published.