utkarshakash said:Homework Statement
cot^-1 3 + cot^-1 7 + cot^-1 13+...
Homework Equations
The Attempt at a Solution
I first tried to write the nth term of the series
[itex]t_n = cot^{-1}\left( 2^n + (2n-1) \right)[/itex]
Then I tried to calculate the limit as n→∞. But I simply can't do that. I mean I don't know how to calculate the limit.
Ray Vickson said:Your presentation is very confusing; use parentheses, like this:
cot^(-1)(3) + cot^(-1)(7) + ... or use "arccot" instead of "cot^-1".
It's not at all obvious to me how 3, 7, 13.. is supposed to be generalised. Do you know that it is intended to be 2n+2n-1, or are you guessing?utkarshakash said:[itex]t_n = cot^{-1}\left( 2^n + (2n-1) \right)[/itex]
An infinite series is a sum of an infinite number of terms. Each term in the series is added to the previous term, and the sum continues indefinitely.
The sum to an infinite series can be found by using the formula S = a / (1-r), where S is the sum, a is the first term, and r is the common ratio between terms. This formula only works for geometric series, which have a constant common ratio between terms.
No, not all infinite series can be summed. Only geometric series and some special cases of other types of series can be summed to a finite value. Other series may diverge, meaning their sum approaches infinity.
A convergent series is one that has a finite sum, meaning the terms in the series eventually get smaller and smaller until they approach zero. A divergent series is one that does not have a finite sum, meaning the terms in the series do not approach zero and the sum approaches infinity.
The sum to an infinite series is used in various mathematical and scientific fields, such as in finance, physics, and engineering. It can be used to calculate compound interest, model population growth, and analyze electrical circuits, among other applications.