Investigate the following for absolute convergence,...

Investigate the following for absolute convergence, conditional convergence, or divergence: (a) $1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots$. Here $\left|s_n\right|=\frac{1}{2^{n-1}},\left|s_{n+1}\right|=\frac{1}{2^n}$, and $R=\lim _{n \rightarrow \infty} \frac{\left|s_{n+1}\right|}{\left|s_n\right|}=\lim _{n \rightarrow \infty} \frac{2^{n-1}}{2^n}=\frac{1}{2}<1$. The series is absolutely convergent. (b) $1-\frac{4}{1 !}+\frac{4^2}{2 !}-\frac{4^3}{3 !}+\cdots$ Here $\left|s_n\right|=\frac{4^{n-1}}{(n-1) !},\left|s_{n+1}\right|=\frac{4^n}{n !}$, and $R=\lim _{n \rightarrow \infty} \frac{4}{n}=0$. The series is absolutely convergent. (c) $\frac{1}{2-\sqrt{2}}-\frac{1}{3-\sqrt{3}}+\frac{1}{4-\sqrt{4}}-\frac{1}{5-\sqrt{5}}+\cdots$. The ratio test fails here. Since $\frac{1}{n+1-\sqrt{n+1}}>\frac{1}{n+2-\sqrt{n+2}}$ and $\lim _{n \rightarrow \infty} \frac{1}{n+1-\sqrt{n+1}}=0$, the series is convergent. Since $\frac{1}{n+1-\sqrt{n+1}}>\frac{1}{n+1}$ for all values of $n$, the series of absolute values is term by term greater than the harmonic series, and thus is divergent. The given series is conditionally convergent.

Investigate the following for absolute convergence, conditional convergence, or divergence:
(a) $1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots$.
Here $\left|s_n\right|=\frac{1}{2^{n-1}},\left|s_{n+1}\right|=\frac{1}{2^n}$, and $R=\lim _{n \rightarrow \infty} \frac{\left|s_{n+1}\right|}{\left|s_n\right|}=\lim _{n \rightarrow \infty} \frac{2^{n-1}}{2^n}=\frac{1}{2}<1$. The series is absolutely convergent.
(b) $1-\frac{4}{1 !}+\frac{4^2}{2 !}-\frac{4^3}{3 !}+\cdots$
Here $\left|s_n\right|=\frac{4^{n-1}}{(n-1) !},\left|s_{n+1}\right|=\frac{4^n}{n !}$, and $R=\lim _{n \rightarrow \infty} \frac{4}{n}=0$. The series is absolutely convergent.
(c) $\frac{1}{2-\sqrt{2}}-\frac{1}{3-\sqrt{3}}+\frac{1}{4-\sqrt{4}}-\frac{1}{5-\sqrt{5}}+\cdots$.
The ratio test fails here.
Since $\frac{1}{n+1-\sqrt{n+1}}>\frac{1}{n+2-\sqrt{n+2}}$ and $\lim _{n \rightarrow \infty} \frac{1}{n+1-\sqrt{n+1}}=0$, the series is convergent.
Since $\frac{1}{n+1-\sqrt{n+1}}>\frac{1}{n+1}$ for all values of $n$, the series of absolute values is term by term greater than the harmonic series, and thus is divergent. The given series is conditionally convergent.

Key Concepts

Comparison Test

The comparison test is a technique for determining the convergence or divergence of a series by comparing it to another series with known behavior. In this method, if the terms of a given series are larger than those of a known divergent series, or smaller than those of a known convergent series (assuming the terms are nonnegative), then similar conclusions can be made for the series in question. This test is a powerful tool when the series under investigation has terms that can be bounded in a meaningful way.

Alternating Series Test

The alternating series test (or Leibniz test) is used to determine the convergence of series whose terms alternate in sign. The test states that if the absolute value of the terms of the series decreases monotonically to zero, then the alternating series converges. However, this test does not imply absolute convergence, and such series may converge only conditionally.

Absolute Convergence

Absolute convergence refers to the situation where the series formed by taking the absolute values of the terms of the original series converges. This concept is central because if a series converges absolutely, then it converges regardless of the signs of its terms. It is often easier to work with absolute values when applying tests for convergence, and absolute convergence implies convergence of the original series.

Conditional Convergence

Conditional convergence occurs when a series converges, yet the series of absolute values diverges. This concept highlights that the convergence of a series can depend delicately on the cancellation between positive and negative terms. Conditional convergence is typical in alternating series where the alternating behavior leads to convergence even if the magnitude of the terms is not sufficiently small to secure absolute convergence.

Ratio Test

The ratio test is a method for determining the absolute convergence of a series by examining the limit of the ratio of successive terms. Specifically, if the limit of |a_(n+1)/a_n| is less than 1, the series converges absolutely; if greater than 1, the series diverges; and if equal to 1, the test is inconclusive. This test is particularly useful for series with factorials, exponentials or combinations of both.

Please give Ace some feedback