Convergence for a series: what is wrong with my method?

[JJHK]

Convergence for a series: what is wrong with my method??

Homework Statement

For the following series, write formulas for the sequence a_n, S_n, and R_n, and find the limits of the sequences as n→∞.

Homework Equations

S_n is the partial sum of the series.

R_n is the remainder and is given by
R_n = S - S_n,
where
lim(n→∞) S_n = S

The Attempt at a Solution

a_n is pretty straightforward...

a_n = Ʃ(1→∞) (-1)ⁿ⁺¹ (2n+1)ⁿ / [ (n) (n+1) ]

and as n→∞, a_n → 0

For S_n, I am getting:

S_n = 3 / (1x2) - 5 / (2x3) + 7 / (3x4) - 9 /(4x5) + ...

= 3 (1/1 - 1/2) - 5 (1/2 - 1/3) + 7(1/3 - 1/4) - 9 (1/4 - 1/5) + ...

= 3 - 1/2 (3+5) + 1/3 (5+7) - 1/4 (7+9) + ...

= 3 - 8/2 + 12/3 -16/4 + ...

= 3 - 4 + 4 - 4 + ...

= 3 + Ʃ(1→∞) (-1)ⁿx4

so as n→∞, does S_n approach 3 or -1, or neither?

Also, what would R_n be?

thank you so much!

 

[LCKurtz]

Homework Statement

For the following series, write formulas for the sequence a_n, S_n, and R_n, and find the limits of the sequences as n→∞.

Homework Equations

S_n is the partial sum of the series.

R_n is the remainder and is given by
R_n = S - S_n,
where
lim(n→∞) S_n = S

The Attempt at a Solution

a_n is pretty straightforward...

a_n = Ʃ(1→∞) (-1)ⁿ⁺¹ (2n+1)ⁿ / [ (n) (n+1) ]

That is not $a_n$. What you have on the right is $S$, assuming the series converges.

and as n→∞, a_n → 0

$a_n$ is the nth term of the series$$
a_n=\frac{-1^n(2n+1)^n}{n(n+1)}$$Unless you have mistyped the problem, I don't think you will find $a_n\rightarrow 0$. In any case you haven't shown any argument.

For S_n, I am getting:

S_n = 3 / (1x2) - 5 / (2x3) + 7 / (3x4) - 9 /(4x5) + ...

= 3 (1/1 - 1/2) - 5 (1/2 - 1/3) + 7(1/3 - 1/4) - 9 (1/4 - 1/5) + ...

= 3 - 1/2 (3+5) + 1/3 (5+7) - 1/4 (7+9) + ...

= 3 - 8/2 + 12/3 -16/4 + ...

= 3 - 4 + 4 - 4 + ...

= 3 + Ʃ(1→∞) (-1)ⁿx4

so as n→∞, does S_n approach 3 or -1, or neither?

Also, what would R_n be?

thank you so much!

$S_n$ is the sum of the first $n$ terms. The expression for $S_n$ would never end with a "...". You can't calculate $S$ or $R_n=S-S_n$ unless the series converges. Go back to the beginning and make sure you copied the problem correctly and start by proving whether or not $a_n\rightarrow 0$.