Compute the finite sum \(\sum_{n=1}^{4} n\).

Find the sum of the infinite geometric series \(\sum_{n=1}^{\infty}3\left(\frac{1}{4}\right)^{n-1}\).

Find the sum of the telescoping series \(\sum_{n=1}^{\infty}\frac{1}{n(n+1)}\).

Compute the fourth partial sum \(S_4\) of the harmonic series \(\sum_{n=1}^{\infty}\frac{1}{n}\).

Find the sum of \(\sum_{n=0}^{\infty}\left(-\frac{1}{2}\right)^n\).

For the series with terms \(a_n=n\), find a formula for the \(N\)th partial sum \(S_N=\sum_{n=1}^{N}a_n\).

Find a formula for \(\sum_{n=1}^{N}2^n\), where \(N\) is a positive integer.

Find the \(N\)th partial sum of \(\sum_{n=1}^{\infty}\ln\left(\frac{n+1}{n}\right)\), and decide whether the series converges.

Find the sum of \(\sum_{n=1}^{\infty}\left(\frac{1}{2n-1}-\frac{1}{2n+1}\right)\).

A learner uses the infinite geometric formula on \(\sum_{n=0}^{\infty}2^n\) and gets \(\frac{1}{1-2}=-1\). Explain the error and state what the series does.