# Determine whether each series converges or diverges. Please show all work for parts A and B. ...

## Question:

Determine whether each series converges or diverges. Please show all work for parts A and B.

(A.) {eq}\Sigma_{n = 1}^{\infty} \frac{\pi}{n ^{\frac{1}{e}}} {/eq}

(B.) {eq}\Sigma_{n = 1}^{\infty} \frac{e}{n^{\pi}} {/eq}

## P-Series:

An infinite series {eq}\sum\limits_{n = 1}^\infty {{u_n}} {/eq} is said to be convergent if the sequence of the nth partial sum is convergent.

An infinite series {eq}\sum\limits_{n = 1}^\infty {\frac{1}{{{n^p}}}} {/eq} is of the form is called p-series.

p-series test:

For an infinite p-series {eq}\sum\limits_{n = 1}^\infty {\frac{1}{{{n^p}}}} {/eq} we have the following discussion,

{eq}\eqalign{ & 1) ~p > 1 \Rightarrow ~\text{series is convergent} \cr & 2) ~p \leqslant 1 \Rightarrow~\text{ series is divergent} \cr} {/eq}

Useful result:

Let, {eq}\sum\limits_{n = 1}^\infty {{u_n}} {/eq} be an infinite series and {eq}\lambda {/eq} be a fixed real number then,

{eq}\sum\limits_{n = 1}^\infty \lambda {u_n} ~\&~ \lambda \sum\limits_{n = 1}^\infty {{u_n}} {/eq}

both convergent or divergent together.

## Answer and Explanation:

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{eq}(A.) \Sigma_{n = 1}^{\infty} \frac{\pi}{n ^{\frac{1}{e}}} {/eq}

Above series can be written as,

{eq}\sum\limits_{n = 1}^\infty {\frac{\pi...

See full answer below.

P-Series: Definition & Examples

from

Chapter 29 / Lesson 5
22K

This lesson is designed to help you understand a specific type of series called a p-series. You will determine if a series is a p-series, and you will learn to decide if a p-series converges or diverges.