MathematicalPhysicist said:Much simpler than that:
[tex]|\frac{1-(-1)^ne}{1+(\pi n)^2}| \leq \frac{1+e}{1+(\pi n)^2}[/tex]
A series converges if the sum of its terms approaches a finite value as the number of terms increases towards infinity. In other words, the series "settles down" to a specific value as more and more terms are added.
There are several methods for proving that a series converges, including the comparison test, the ratio test, and the root test. These tests involve comparing the series to other known convergent or divergent series and using mathematical manipulations to determine the convergence or divergence of the original series.
No, a series can only converge to a single value. If a series converges, it means that the terms are "settling down" to a specific value and will not change. If a series appears to converge to multiple values, it is likely that the series is divergent.
If a series does not converge, it is said to be divergent. This means that the sum of the terms does not approach a finite value as the number of terms increases towards infinity. Divergent series can either have an infinite sum or oscillate between different values.
One example of a convergent series is the geometric series 1/2 + 1/4 + 1/8 + 1/16 + ..., which converges to a sum of 1. This can be proven using the ratio test, as the ratio between consecutive terms approaches 1/2 as the number of terms increases towards infinity.