For \(\sum_{n=1}^{\infty}a_n\), what is \(S_N\)?

State the term test condition that every convergent series must satisfy.

Find \(S_3\) for \(\sum_{n=1}^{\infty}1/n\).

Use the term condition on \(\sum_{n=1}^{\infty}\frac{n}{n+1}\).

Find \(S_1,S_2,S_3,S_4\) for the series \(1+2+3+4+\cdots\).

For \(a_n=(-1)^n\), find the first four partial sums of \(\sum_{n=1}^{\infty}a_n\).

If \(S_N=3-2/N\), find the sum of the series if these are its partial sums.

If \(S_N=N/(N+2)\), decide whether the corresponding series converges and find the limiting sum.

Use partial sums to show that \(\sum_{n=1}^{\infty}n\) diverges.

A series has partial sums \(S_N=5+(-1)^N/N\). Determine its convergence and sum.

Explain why \(a_n\to0\) does not prove convergence, using \(a_n=1/n\).

For \(\sum_{n=0}^{\infty}a_n\), write \(S_N\) and state how many terms it contains.

Find \(\lim_{N\to\infty}\frac{4N+1}{2N-3}\) and interpret it as a series sum.

If \(S_N=(-1)^N+1/N\), does the corresponding series converge?

Given \(S_N=\sum_{n=1}^{N}(2n-1)\), show the infinite series diverges.

If \(S_N=L+c/(N+1)\), where \(L\) and \(c\) are constants, prove convergence and find the sum.

Let \(S_N=1-1/(N+1)\) and \(S_0=0\). Find \(a_1\) and \(a_2\) using \(a_N=S_N-S_{N-1}\).

Diagnose the error: \(a_n\to0\), therefore \(\sum a_n\) converges.

Prove that if \(S_N\to S\), then the terms \(a_N\) of the series tend to \(0\).

A model has partial sums \(S_N=10-3(0.8)^N\). Explain why the approximations stabilize and find the limiting total.