Power Series [n=2,infty][ {(n-1)(-1)^(n)} / n! ] Convergence Assistance

[testing123]

Power Series [n=0,infty][ {(2n-1)(-1)^(n)} / n! ] Convergence Assistance

(2n-1)(-1)^n/(n-1!)

 

[tkhunny]

1) Does it converge at all?
2) If so, please demonstrate your best efforts at a determination of the limit.

 

[testing123]

1) Does it converge at all?
2) If so, please demonstrate your best efforts at a determination of the limit.

Using the Ratio Test, I calculated that the limit as n approaches infinity equals 0. Since L<1, the power series converges by the Ratio Test. Does this mean that the power series converges to 0?

 

[tkhunny]

NO, only that the individual TERMS approach zero.
The sum of the series is a different matter.


1 + 1/2 + 1/4 + 1/8 + ...

The TERMS approach zero (0).
The SERIES approaches 2.

 

[testing123]

NO, only that the individual TERMS approach zero.
The sum of the series is a different matter.


1 + 1/2 + 1/4 + 1/8 + ...

The TERMS approach zero (0).
The SERIES approaches 2.

Would reindexing the problem and then taking the ratio test work?

 

[greg1313]

NO, only that the individual TERMS approach zero.
The sum of the series is a different matter.


1 + 1/2 + 1/4 + 1/8 + ...

The TERMS approach zero (0).
The SERIES approaches 2.

The partial sums of 1, 1 + 1/2, 1 + 1/2 + 1/4, etc., approach 2.

The sum of the series does not approach 2. It is 2.

 

[tkhunny]

The partial sums of 1, 1 + 1/2, 1 + 1/2 + 1/4, etc., approach 2.

The sum of the series does not approach 2. It is 2.

Fair enough.

 

[tkhunny]

Would reindexing the problem and then taking the ratio test work?

Demonstrating convergence is one thing.
--- The terms approaching zero is one piece of this.
--- The limit of the sequence of partial sums approaching something is another part of this. <== The Ratio Test often will help with this.

Demonstrating the Actual Limit of the Sequence of Partial sums (the sum of the series) is quite another matter. <== No help from the Ratio Test, here.