Can the Partial Sum of the Cosine Telescoping Series be Negative?

[Jbreezy]

Homework Statement

If you sum this
from one to infinity.
Ʃ (cos(1/(n)^2 - cos(1/(n+1)^2)

Homework Equations

The Attempt at a Solution

Ʃ (cos(1/(n-1)^2 - cos(1/(n+1)^2)
This is telescoping if you work that out for the partial nth partial sum you get

cos(1) - cos(1/(n+1)^2) if you take the limit you get cos(1) -1 which is negative. If you punch it in on wolfram you get the same thing. Can the partial sum be negative ?

 

[LCKurtz]

Homework Statement

If you sum this
from one to infinity.
Ʃ (cos(1/(n)^2 - cos(1/(n+1)^2)

Homework Equations

The Attempt at a Solution

Ʃ (cos(1/(n-1)^2 - cos(1/(n+1)^2)
This is telescoping if you work that out for the partial nth partial sum you get

cos(1) - cos(1/(n+1)^2) if you take the limit you get cos(1) -1 which is negative. If you punch it in on wolfram you get the same thing. Can the partial sum be negative ?

Yes.

 

[Jbreezy]

Can you elaborate please.

 

[LCKurtz]

What's to elaborate? Partial sums can be negative. You are looking at one. Plug in some values of n and see.