Alternating series

The alternating-series test provides a simple way to estimate the truncation error: the absolute value of the remainder $|R_n|$ after summing $n$ terms is bounded by the absolute value of the very next term, $\frac{1}{n+1}$.

Conditions: The series satisfies the conditions of the alternating-series test.; The error is estimated after summing $n$ terms.

The alternating-series test provides sufficient conditions for convergence: the absolute value of the general term must be monotonically decreasing, and the term itself must tend to zero. If these conditions are met, the series converges.

Conditions: The series is an alternating series.; The absolute values of the terms are nonnegative.; The sequence of absolute values is monotonically decreasing.; The limit of the terms as n approaches infinity is zero.

The partial sums of the alternating harmonic series oscillate around the limit $\ln(2)$. Odd partial sums approach the limit from one side, and even partial sums approach it from the opposite side, getting closer and closer with each additional term.

Conditions: The series is the alternating harmonic series $\sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{n}$.; The limit of the series is $\ln(2)$.

A finite plot of partial sums only shows the behavior of the series up to a certain number of terms. It cannot guarantee that the sequence of partial sums will continue to approach a limit indefinitely.

Conditions: The plot shows only a finite number of partial sums.; Convergence is defined as the limit of the infinite sequence of partial sums.

Monotonicity is a sufficient condition for convergence under the alternating-series test, but it is not necessary. Some alternating series can converge even if their terms do not decrease monotonically, as long as the terms still tend to zero and the overall sum stabilizes.

Conditions: The series is an alternating series.; The terms tend to zero as n approaches infinity.