Absolute convergence of a series. Hello friends, today I’ll talk about the absolute convergence of a series. Have a look!!

**Absolute convergence of a series**

Now for any general series, *the condition for absolute convergence* is:

- If converges, is absolutely convergent.
- If diverges but converges, is conditionally convergent.

And in order to test the convergence of any series, I’ll use D’ Alembert’s ratio test for positive terms.

The outcomes of this test are as follows:

Now I’ll give some examples on the absolute convergence of a series.

**Solved examples of the absolute convergence of a series**

Disclaimer: None of these examples are mine. I have chosen these from some book or books. Also I have given the due reference at the end of the post. I only own the solution of these examples.

So here is the first example.

**Example 1**

According to Stroud and Booth (2013), “Find the range of values of for which the series

is absolutely convergent.”

**Solution**

Now here the given series is

And I have to find out the range of values of for which this series is absolutely convergent. So I’ll check if converges or not. Therefore I’ll start with the D’ Alembert’s ratio test.

**Step 1**

First of all, I’ll get the value of .

As I know, the th term of the series is

So the th term will be

And that means

Therefore the value of will be

Since , I can say that

Now will be convergent if

Next, I’ll get the limiting value of when tends to infinity.

Related post: **How to get the limit of a series**

**Step 2**

So that means

Since is not a function of , I can take it out. So the limit will be

Now is the highest power of in this expression. So I’ll divide both the numerator and the denominator with . And that gives

Since tends to infinity, tends to .

So the limit becomes

Now for the absolute convergence of any series,

So in this case, this means

Therefore for , the series is absolutely convergent. 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), “show that the series

is convergent if and for no other values of .”

**Solution**

Now here the given series is:

And I have to prove that this series is convergent if and for no other values of . First of all, I’ll get the th term of the series.

**Step 1**

So I can rewrite the series as

Hence with the th term, the series will be

Therefore the th term is

Thus the th term is

So this means

Now I’ll check if converges or not. Therefore I’ll start with the D’ Alembert’s ratio test.

**Step 2**

First of all, I’ll get the value of . And that means

Now I’ll simplify it. And that gives

Since both and are not equal to , I can say that the value of is

Next, I’ll separate the terms. So it will be

Now will be convergent if

Next, I’ll get the limiting value of when tends to infinity.

**Step 3**

So that means

Since is not a function of , I can take it out. So the limit will be

Now is the highest power of in this expression. So I’ll divide both the numerator and the denominator with . And that gives

Since tends to infinity, tends to .

So the limit becomes

**Step 4**

Now for the absolute convergence of any series,

So in this case, this means

And that gives

Also, for the absolute divergence of any series,

So in this case, this means

And that gives

Thus for or for , the given series is divergent. Therefore the series is only convergent for . Thus I have proved that this series is convergent if and for no other values of . And this is the answer to the given 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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