Jbreezy said: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 ?
A partial sum is the sum of a finite number of terms in a sequence. It is the sum of the first n terms in a sequence, where n is any positive integer.
A partial sum can be negative if the terms in the sequence alternate between positive and negative values, resulting in a net negative sum. For example, the partial sum of the sequence 1, -2, 3, -4, 5 would be -2.
If a partial sum is negative, it means that the sum of the first n terms in the sequence is less than zero. This could indicate that the overall trend of the sequence is decreasing or that the terms in the sequence are alternating between positive and negative values.
No, a partial sum cannot be negative for all sequences. Some sequences may have only positive terms and therefore the partial sum would always be positive. Additionally, the partial sum of an infinite sequence can approach zero but never be negative.
The partial sum of a sequence is calculated by adding up the first n terms in the sequence, where n is any positive integer. This can be done by hand or using a calculator or computer program.