# Absolute convergence help

miglo

## Homework Statement

$$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{\sqrt{n}+1}{n+1}$$

## Homework Equations

absolute convergence test

## The Attempt at a Solution

by book says that the series converges because $$\sum_{n=1}^{\infty}\frac{\sqrt{n}+1}{n+1}$$ converges
but they don't show how the absolute value of the original series converges, and I've tried showing it myself but i keep getting divergence
i know that as n grows larger and larger the behavior of $$\frac{\sqrt{n}+1}{n+1}$$ is similar to that of $$\frac{\sqrt{n}}{n}$$ so i tried using limit comparison and direct comparison with $\frac{1}{n}$ but i keep getting divergence
i tried the integral test but i kept getting divergence also
ive been trying this for far too long so any help would be greatly appreciated

Homework Helper
Gold Member
You are correct that the positive term series diverges.

miglo
but my book says that the original series converges by the absolute convergence test
so wouldn't that mean that $$\sum_{n=1}^{\infty}\frac{\sqrt{n}+1}{n+1}$$ converges also? or is this an error in the book?

so wouldn't that mean that $$\sum_{n=1}^{\infty}\frac{\sqrt{n}+1}{n+1}$$ converges also? or is this an error in the book?