We already learned about arithmetic, geometric as well as telescoping series. Today let’s learn how to evaluate series. If you’re thinking to create your own formula, this is the perfect topic for you. Before I teach you to create your own formula and you’re so lucky if this is your first time. We need to know first some basic information. One of them is to know what summation symbol is all about.

Short Hand Expanded

**Foundation Formula:**

(1)

(2)

(3)

For all positive integer *a* and

(4)

(5)

(6)

Some Rules to follow:

where is constant and is a function. (i)

(ii)

**Sample Problem 1:**

Solve the formula for the sum of

Step 1: Write the expression to its shorthand notation using the summation symbol

Step 2: Evaluate the series following the rules and use the basic foundation of series

But and

Thus,

**Sample Problem 2:**

Find the sum of series

**Solution:**

We need to find the first the formula for the nth term of the series. It is obvious that the nth term can be express as .

In shorthand notation we can express the sum as

Simplifying the notation we have

Using rule (i) & (ii)

Using formula (3),(2), and (1) we have

But is a common factor

By factoring,

Hence,

Going back to the problem we have

** **

**Sample Problem 3:**

Find the sum of the infinite series

Expressing the sum in shorthand notation we have

This notation is similar to formula (5) with

** **

**Sample Problem 4:**

Find the sum of infinite series

**Solution: **We can express the series in shorthand notation as follows.

The notation above is similar to formula (6) where

**Sample Problem 5:**

Find the sum of infinite series

**Solution:** We can express the sum in shorthand notation as follows

Using the rule (ii)

The two sums are similar to (6) and (5) respectively where