- #1
kylera
- 40
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Alternating Series and P-series "convergence"
I couldn't resist trying out a pun. Anyway, onto the question:
Test the series for convergence/divergence:
[tex]\sum^{\infty}_{n=1}\frac{-1^{n-1}}{\sqrt{n}}[/tex]
Alternating Series Test and possibly p-series test...
The expression for an alternating series goes as a(n) = (-1)^(n-1) * b(n). Having said that, it's obvious that b(n) is
[tex]\frac{1}{\sqrt{n}}[/tex]
which can be re-written as [tex]\frac{1}{n^\frac{1}{2}}[/tex]. But by the rules of the p-series, since 0.5 is obviously lesser than 1, I came to the conclusion that the series diverges. Instead, the answer states that the series does indeed converge. Can anyone help shed some light in this?
I couldn't resist trying out a pun. Anyway, onto the question:
Homework Statement
Test the series for convergence/divergence:
[tex]\sum^{\infty}_{n=1}\frac{-1^{n-1}}{\sqrt{n}}[/tex]
Homework Equations
Alternating Series Test and possibly p-series test...
The Attempt at a Solution
The expression for an alternating series goes as a(n) = (-1)^(n-1) * b(n). Having said that, it's obvious that b(n) is
[tex]\frac{1}{\sqrt{n}}[/tex]
which can be re-written as [tex]\frac{1}{n^\frac{1}{2}}[/tex]. But by the rules of the p-series, since 0.5 is obviously lesser than 1, I came to the conclusion that the series diverges. Instead, the answer states that the series does indeed converge. Can anyone help shed some light in this?