Unit 4A Practice Exam B Answer Key

It is increasing and bounded above by $3$; a positive limit satisfies $L^2-L-3=0$, hence $L=(1+\sqrt{13})/2$.

First term $2/3$, ratio $1/3$, sum $1$.

Converges by the Integral Test.

$R_N\le\int_N^\infty x^{-4}dx=1/(3N^3)$.

Converges by limit comparison with $1/n^2$.

Diverges because terms do not approach zero.

Alternating convergence; absolute series behaves like $1/(n\ln n)$ and diverges, so conditional.

Ratio limit $1/2$; converges absolutely.

The nth root is $1/n^{7/n}\to1$, so the test gives no conclusion.

Example: $\sum_{n=1}^{\infty}(-1)^{n-1}/n$. It converges by the Alternating Series Test, while the absolute series is harmonic and diverges; therefore convergence is conditional.

For every $\varepsilon>0$, there is $N$ such that whenever $n>m\ge N$, $|a_{m+1}+\cdots+a_n|<\varepsilon$. Thus all sufficiently late partial sums are mutually close.

Any finite sample is compatible with many possible later behaviors.