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 is the common difference.

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

For

Now here is the first term and is the common ratio.

###### 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 terms, the series .”

**Solution**

Now here the given series is:

In order to find the sum to terms, first of all, I’ll get the th term of the series. So I’ll start with the first element of the th term.

**Step 1**

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

So this is an arithmetic series with the first term as and the common difference as . Let’s say is the first element of the th term.

Thus as per the standard formula in series,

(1)

Similarly, I’ll get the th term of the second and third elements as well.

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

So this is an arithmetic series with the first term as and the common difference as . Let’s say is the first element of the th 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 as and the common difference as . Let’s say is the first element of the th term.

Thus as per the standard formula in series,

(3)

**Step 2**

Hence I can say that the th term of the given series is

Next, I’ll get the values of and from equations (1) – (3) to get the value of as

Then I’ll simplify it to get

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

(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 find the th term and the sum of the first terms.”

**Solution**

Let the given series is

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

First of all, I’ll rewrite this series so that I can get its th term, say, .

**Step 1**

As we all know, means . Similarly, means . So I can say that the series is

Also, I know that is and is . Thus the series can be

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

As we all know , with as a natural number. Similarly, I can say that and . Therefore the series will be

Now I will rewrite this series as

Thus the th term of the series will be

(5)

Now I’ll get the sixth term of the series. So I’ll substitute in equation (5).

Thus the sixth term of the series will be

Next, I’ll get the sum of the first terms of the series .

**Step 2**

As we all know, the sum of terms of a geometric series is

where is the first term and is the common ratio. Now here the given series is also a geometric series with the first term as . And the common ration is .

So the sum of this series of terms is

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 and the sum of its first terms is . Hence I can conclude that these are the answers to this example.

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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