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

**Convergence of series**

There are several ways to test the convergence of a series as I have already mentioned in my * collection of formulas in series*.

So these are as follows.

*Test 1*

If , the series may be convergent.

If , the series is divergent.

*Test 2*

It works for the standard series.

Suppose the series is:

Now this series is convergent for . And for , the series is divergent.

**Test 3 (D’ Alembert’s ratio test for positive terms)**

If , the series converges.

If , the series diverges.

And for , the test doesn’t work.

**Examples of convergence of 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)* “Determine whether the following series is convergent: .”

**Solution**

Now here the given series is . So to check if it’s convergent or not, I’ll use Test 1 first. If that doesn’t work, then I’ll try Test 3. Since this is not a standard series, I can’t use Test 2 for it.

So I’ll get the * limit of the series* when tends to infinity. Thus it will be

Since is the highest power of in , I’ll divide both the top and bottom of with . And that gives

Now for . So I can say

Thus the limit of when is . And that is a distinct number. So as per Test 1, the series is divergent.

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)* “Determine whether the following series is convergent: .”

**Solution**

Now here the given series is . So to check if it’s convergent or not, I’ll use D’ Alembert’s ratio test in Test 3 here. Since this is not a standard series, I can’t use Test 2 for it.

So will be

Hence is

So I can say that

Now I’ll get the limit of

Since is the highest power of in , I’ll divide both the numerator and denominator with . And that gives

Now for tends to infinity, tends to be .

Thus I can say that

which is less than . So the series converges.

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

Now I’ll give another example.

**Example 3**

According to Stroud and Booth (2013)* “Prove that

and that

”

**Solution**

Now here the first series is

Also I can rewrite it as

As I can see, it’s a standard series. So I can use the Test 2 to check its convergence. As per Test 2, if the power is less than or equals to , the series diverges. Now I’ll compare it with the standard series

So this gives which is less than . So I can say that this series is divergent.

Hence I have proved it.

Now I’ll check the second series

As I can see, this is also a series of natural numbers with the power . Since , the series is convergent.

Thus I have proved it as well.

Hence I can conclude that I have proved both relations. And this is the answer to the given example.

Now I’ll give another example.

**Example 4**

According to Stroud and Booth (2013)* “Show that the following series is convergent: .”

**Solution**

So here the given series is:

Now I’ll rewrite the series so that I can write the nth term of the series. Thus it will be

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

Thus to check if it’s convergent or not, I’ll use D’ Alembert’s ratio test in Test 3 here. Since this is not a standard series, I can’t use Test 2 for it.

Hence the th term of the series is

And this gives

Hence is

Now I’ll get the limit of

Since is the highest power of in , I’ll divide both the numerator and denominator with . And that gives

Now for tends to infinity, tends to be , tends to be .

Thus I can say that

And this gives

which is less than . So the given series converges.

Hence I can conclude that 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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