I am unsure of which test to use. At first, I thought I should use alternating series test, but I am unable to manipulate the numerator to

What series test would I use, and how?

Teddy Dillard
2021-12-25
Answered

I am having difficulty finding if this series converges or diverges:

$\sum _{n=0}^{\mathrm{\infty}}\frac{{(-2)}^{n+1}}{{n}^{n-1}}$

I am unsure of which test to use. At first, I thought I should use alternating series test, but I am unable to manipulate the numerator to$(-1)}^{n+1$ . I thought I may be able to use root test, but I have no clue how to manipulate the series to do that.

What series test would I use, and how?

I am unsure of which test to use. At first, I thought I should use alternating series test, but I am unable to manipulate the numerator to

What series test would I use, and how?

You can still ask an expert for help

Ethan Sanders

Answered 2021-12-26
Author has **35** answers

Alternating series test

Note that

$\sum _{n=0}^{\mathrm{\infty}}\frac{{(-2)}^{n+1}}{{n}^{n-1}}=\sum _{n=0}^{\mathrm{\infty}}{(-1)}^{n+1}\frac{{2}^{n+1}}{{n}^{n-1}}$

By the alternating series test this converges if

1.${a}_{n}>0$ : we can easily this is satisfied

2.$a}_{n$ is monotic: $a}_{n$ is decreasing monotonically

3.$\underset{n\to \mathrm{\infty}}{lim}{a}_{n}=0$ I will leave this one to you.

Comparison test

$\left|\sum _{n=0}^{\mathrm{\infty}}\frac{{(-2)}^{n+1}}{{n}^{n-1}}\right|\le \sum _{n=0}^{\mathrm{\infty}}\frac{{2}^{n+1}}{{n}^{n-1}}\le \sum _{n=0}^{3}{\left(\frac{2}{n}\right)}^{n+1}+\sum _{n=4}^{\mathrm{\infty}}{\left(\frac{2}{n}\right)}^{n+1}$

The last series is a geometric series and therefore converges.

Note that

By the alternating series test this converges if

1.

2.

3.

Comparison test

The last series is a geometric series and therefore converges.

Virginia Palmer

Answered 2021-12-27
Author has **27** answers

Hint: what can you say about $\frac{{2}^{n-1}}{{n}^{n-1}}$ ?

Can you show that$\frac{{2}^{n-1}}{{n}^{n-1}}\le \frac{1}{{n}^{2}}$ for $n\ge N$ ?

Can you show that

user_27qwe

Answered 2021-12-30
Author has **208** answers

Making the problem more general, let

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