the convergence of a series

ffrogg

Prove that the series Σ{ （-1）Λ[sqrt(n)]}/n converges. []stands for the Gauss's function.
Please tell me your opinion.Thank you!

**Posted:** Sun Feb 13, 2005 4:39 am

Nené · Poincare Conjecture Offline Joined: 19 Jan 2005 Posts: 170

We can follow this line of reasoning:
the series $\sum \frac {-1^{[\sqrt {n}]}} {n^a}$
converges absolutely if $\ a > 1 \$ and diverges if $\ a \leq 0$ . We will show that if $\frac {1} {2} < a \leq 1$ the series converges conditionally. Observe that the first three terms of the series are negative, the next five terms are positive, etc. Now grouping the terms of the same sign we get the alternating series $\displaystyle \sum_ {n=1}^{\infty} (-1)^n C_n \ where \ C_n = \sum_{k=n^2}^{(n+1)^2-1} \frac {1} {k^a}$ .
Now if $C_n - C_{n + 1} > 0$ the series for the theorem of Leibniz converges.

$C_n - C_{n + 1} = \sum_{k=n^2}^{(n+1)^2-1} \frac {1} {k^a} - \sum_{k=(n+1)^2}^{(n+2)^2-1} \frac {1} {k^a}= \sum_{k=n^2}^{(n+1...$
where the last inequality follow from the montonicity of the function:
$g(x) = \frac {1} {(n^2+x)^a} - \frac {1} {((n+1)^2+x)^a}$
on the interval[0,2n]. For sufficiently large n:
$C_n - C_{n + 1} > 0$

ffrogg

According to the book,the series converges when a>1/2 and diverges when a<1/2.
So there may be some mistakes in Nené's statements.
Please tell me your opinion.Thank you!

**Posted:** Sun Feb 13, 2005 10:16 pm

Nené · Poincare Conjecture Offline Joined: 19 Jan 2005 Posts: 170

O.K. ffrogg.
The series, if $0 < a \leq \frac {1} {2}$ , dosn't converge. Since $C_n > (2n+1) \frac {1} {(n^2+2n)^a}$ , the necessary condition for the convergence of series isn't satisfied.

**Posted:** Mon Feb 14, 2005 2:12 am