LaTex and solution for an infinite series

[Loren Booda]

What is the LaTex and infinite sum for 1-2^-1/2+3^-1/2-4^-1/2+5^-1/2 . . .

Does it converge anyway?

I am too old for this to be a school assignment.

 

[DonAntonio]

What is the LaTex and infinite sum for 1-2^-1/2+3^-1/2-4^-1/2+5^-1/2 . . .

Does it converge anyway?

I am too old for this to be a school assignment.

A PF contributor that doesn't know LaTeX? Strange...anyway: $$1-2^{-1/2}+3^{-1/2}-4^{-1/2}+...=1-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}-\frac{1}{\sqrt{4}}+...=\sum_{n=1}^\infty\frac{(-1)^{n-1}}{\sqrt{n}}$$

The sum converges as it is an alternating Leibnitz series: the general term sequence converges monotonically to zero and we have alternating signs.

DonAntonio

 

[micromass]

A LaTeX guide is here: https://www.physicsforums.com/showthread.php?t=546968

 

[Xitami]

http://en.wikipedia.org/wiki/Dirichlet_eta_function

 

[alphy]

LaTex is a typesetting language used in mathematics and science to write and format complex equations and formulas. It is widely used in scientific research and publications.

The given infinite series is known as the Alternating Harmonic Series and it can be represented in LaTex as:

$$ \sum_{n=1}^{\infty} (-1)^{n+1}\frac{n}{2} $$

To determine if this series converges, we can use the Alternating Series Test. This test states that if the terms of an alternating series decrease in absolute value and approach zero, then the series converges.

In this case, the terms do decrease in absolute value as n increases and approach zero. Therefore, the series converges to a finite value. This can also be verified by using the Ratio Test or the Comparison Test.

In conclusion, the LaTex representation for the given infinite series is $$ \sum_{n=1}^{\infty} (-1)^{n+1}\frac{n}{2} $$ and it does converge to a finite value.