91Ó°ÊÓ

What is a geometric series? What determines the convergence of a geometric series?

Fill in the blanks to complete each of the following theorem statements.

Basic Limit Rules for Convergent Sequences: If $\left\{a_k\right\}\mathrm{and}\left\{b_k\right\}\mathrm{are}\mathrm{convergent}\mathrm{sequences}\mathrm{with}{a}_k→L\mathrm{and}{b}_k→M\mathrm{as}k→∞$and if c is any constant, then

Ifrole="math" localid="1649231795638" $a_k→L\mathrm{and}{a}_k→M\mathrm{then}\mathrm{\_\_\_\_\_\_\_}.$

In Exercises 4–11, give examples of sequences satisfying the given conditions or explain why such an example cannot exist.
Two divergent sequences $\left\{a_k\right\}$and $\left\{b_k\right\}$such that the sequence $\left\{\frac{a_k}{b_k}\right\}$diverges.

Prove that the ratio of successive terms of a nonzero geometric sequence is constant

Prove that a sequence $\left\{a_k\right\}$that is both increasing and decreasing is constant.

Prove that every sequence of the form ${{\left\{a_k\right\}}_{k=n}}^{∞}$can be rewritten as a sequence of the form ${{\left\{a_k\right\}}_{k=1}}^{∞}$.

Prove that if${{\left\{a_k\right\}}_{k=1}}^{∞}$ is a sequence of positive real numbers, then the sequence ${{\left\{S_n\right\}}_{n=1}}^{∞}$, where the sequence Sn = a1 + a2 +···+an, is an increasing sequence.

Let $\left\{a_k\right\}$be a sequence. Prove Theorem 7.6 (a) along with the following variations:

(a) Show that when role="math" localid="1649277359535" $a_{k-1}-a_k$≥ 0 for every k ≥ 1, the sequence is increasing.

(b) Show that when $a_{k-1}-a_k$> 0 for every k ≥ 1, the sequence is strictly increasing.

(c) Show that when role="math" localid="1649277346412" $a_{k-1}-a_k$≤ 0 for every k ≥ 1, the sequence is decreasing.

(d) Show that when $a_{k-1}-a_k$< 0 for every k ≥ 1, the sequence is strictly decreasing.

Let $\left\{a_k\right\}$be a sequence of positive terms. Prove Theorem 7.6 (b) along with the following variations:

(a) Show that when $\frac{a_{k+1}}{a_k}â‰\yen 1$≥ 1 for every k ≥ 1, the sequence is increasing.

(b) Show that when localid="1649277252156" $\frac{a_{k+1}}{a_k}>1$for every k ≥ 1, the sequence is strictly increasing.

(c) Show that when localid="1649277261045" $\frac{a_{k+1}}{a_k}≤1$for every k ≥ 1, the sequence is decreasing.

(d) Show that when localid="1649277269413" $\frac{a_{k+1}}{a_k}<1$for every k ≥ 1, the sequence is strictly decreasing.

Let a(x) be a differentiable function on the interval [1,∞), and let a_k = a(k) for every positive integer k. Prove Theorem 7.6 (c) along with the following variations:

(a) Show that when a^'(x) ≥ 0f or x > 1,thesequence $\left\{a_k\right\}$is increasing.

(b) Show that when a^'(x) > 0 for x > 1,thesequence $\left\{a_k\right\}$is strictly increasing.

(c) Show that when a'(x) ≤ 0 for x > 1,thesequence $\left\{a_k\right\}$is decreasing.

(d) Show that when a'(x) < 0, for x > 1, the sequence $\left\{a_k\right\}$is strictly decreasing.