Is the Series Convergent or Divergent?

[Ethan Godden]

Homework Statement

I am supposed to determine whether the summation attached is convergent or divergent

Homework Equations

Alternating Series Test
Test for Divergence

The Attempt at a Solution

The attempted solution is attached. Using the two different tests I am getting two different answers.

 

[LCKurtz]

It is much preferred for you to type the problems rather than post a download.

You have $\frac 1 {\sqrt{n+1}}\to 0$ which is correct. Now since$$
0 \le \left | \frac {(-1)^n} {\sqrt{n+1}}\right | \le \frac 1 {\sqrt{n+1}}$$ how could the alternating one not go to zero? And, by the way, $(-1)^\infty$ makes no sense.

 

[Ethan Godden]

Okay, you used the squeeze theorem which makes sense, but why doesn't the test for divergence work? Isn't (-1)^∞ undefined meaning the limit is undefined meaning the series is divergent?

 

[LCKurtz]

Okay, you used the squeeze theorem which makes sense, but why doesn't the test for divergence work? Isn't (-1)^∞ undefined meaning the limit is undefined meaning the series is divergent?

Yes, as I said, $(-1)^\infty$ makes no sense or, as you say, is undefined. What is happening in this problem is that the denominator is getting larger and the numerator is either plus or minus 1 for any n. The fraction gets small no matter the sign, so regardless of the alternating sign the fraction goes to zero.