Sum of a series. Hello friends, today it’s all about the sum of a series. Have a look!!

**Sum of a series**

Here I’ll use some * basic formulas in series* to get the sum of any series. These are as follows:

###### Sum of any arithmetic series:

Now here is the first term and

###### Sum of any geometric series:

For

Now here

###### Sum of any series of natural numbers

For

For

For

Now I’ll give some examples.

**Solved examples of the sum of a series**

Note: None of these examples are mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.

So here is the first example.

**Example 1**

According to Stroud and Booth (2013), “Sum to

**Solution**

Now here the given series is:

In order to find the sum to

**Step 1**

As I can see, here the first elements of the series are

So this is an arithmetic series with the first term

Thus as per the standard formula in series,

(1)

Similarly, I’ll get the

As I can see, here the second elements of the series are

So this is an arithmetic series with the first term

Thus as per the standard formula in series,

(2)

Now the third elements of the series are

So this is an arithmetic series with the first term

Thus as per the standard formula in series,

(3)

**Step 2**

Hence I can say that the

Next, I’ll get the values of

Then I’ll simplify it to get

Therefore I can say that the

(4)

Thus the sum of the series is

So this means

**Step 3**

As I know from the standard formulas in series, the sums of the powers of the natural numbers are as follows:

For

For

And for

So the sum of the given series is

Then I’ll simplify it to get

Thus I can say that the sum of the given series is

Hence I can conclude that this is the answer to the given example.

Now I’ll give another example.

**Example 2**

According to Stroud and Booth (2013), “For the series

**Solution**

Let the given series is

And I have to find its sixth term and the sum of the first

First of all, I’ll rewrite this series so that I can get its

**Step 1**

As we all know,

Also, I know that

Next, I’ll write this series as a power of

As we all know

Now I will rewrite this series as

Thus the

(5)

Now I’ll get the sixth term of the series. So I’ll substitute

Thus the sixth term

Next, I’ll get the sum of the first

**Step 2**

As we all know, the sum of

where

So the sum of this series of

Thus the sum of the series for the first ten terms will be

Therefore I can say that the sixth term of the series is

Dear friends, this is the end of today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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