W07 Regular

Due date: Sunday 3/8, 11:59pm

Sequences

01

03

Computing the terms of a sequence

Calculate the first four terms of each sequence from the given general term, starting at $n=1$:

(a) $\cos \pi n$ $$ (b) ${\displaystyle \frac{n!}{2^n}}$ $$ (c) ${(-1)}^{n+1}$ $$ (d) ${\displaystyle \frac{n}{n+1}}$ $$ (e) ${\displaystyle \frac{3^n}{n!}}$ $$ (f) ${\displaystyle \frac{(2n-1)!}{n!}}$

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Solution

Solutions---0140-03

02 $★$

02

Squeeze theorem

Determine whether the sequence converges, and if it does, find its limit:

(a) $a_n={\displaystyle \frac{{\cos }^{2}n}{2^n}}$ $$ (b) $b_n={\displaystyle {(2^n+3^n)}^{1/n}}$

(Hint for (b): Verify that $3\le b_n\le {(2\cdot 3^n)}^{1/n}$.)

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Solution

Solutions---0140-02

Series

03 $★$

05

Series from its partial sums

Suppose we know that the partial sums $S_N$ of a series $S={\sum }_{n=1}^{\infty }{a}_n$ are given by the formula $S_N=5-\frac{2}{N^2}$.

(a) Compute $a_3$.

(b) Find a formula for the general term $a_n$.

(c) Find the sum $S$.

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Solution

Solutions---0150-05

04 $★$

06

Partial sums and total sum

Consider the series:

$$\sum _{n=1}^{\infty }\frac{{(-8)}^{n-1}}{9^n}$$

(a) Compute a formula for the $N^{\text{th}}$ partial sum $S_N$ by applying the “shift method” steps using the values in this series.

(b) By taking the limit of this formula as $N\to \infty$, find the value of the series.

(c) Find the same value of the series by computing $a_0$ and $r$ and plugging into $S=\frac{a_0}{1-r}$.

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Solution

Solutions---0150-06