Exercises: Series

$\newenvironment {prompt}{}{} \newcommand {\ungraded }[0]{} \newcommand {\bigmath }[1]{$\displaystyle #1$} \newcommand {\choicebreak }[0]{} \newcommand {\type }[0]{} \newcommand {\notes }[0]{} \newcommand {\keywords }[0]{} \newcommand {\offline }[0]{} \newcommand {\comments }[0]{\begin {feedback}} \newcommand {\multiplechoice }[0]{\begin {multipleChoice}} \newcommand {\HyperFirstAtBeginDocument }[0]{\AtBeginDocument }$

Exercises relating to fundamental properties of series.

Compute the sum of the series:

Split the series into two separate geometric series and evaluate each separately.

$\sum _{n=0}^\infty \frac {3^n - 2^n}{4^n} = \answer {2}.$

Compute the sum of the series:

One way to proceed is to reindex the series so that it starts at $n = 0$ .

$\sum _{n=3}^\infty \frac {(-1)^{n+1} 2^n}{3^n} = \answer { \frac {8}{45} }$

Compute the sum of the series: $\sum _{n=1}^\infty \frac {3^{2n+1}}{2^{3n} \cdot 9^n} = \answer {\frac {3}{7}}$

Simplify as much as possible and write in standard form for a geometric series.

Give an example of a geometric series whose first term is $3$ and which sums to $6$ . If no such series exists, enter N/A in both spaces below: $6 = \sum _{n=0}^\infty \answer {3} \left ( \answer {\frac {1}{2}} \right )^n$
Give an example of a geometric series whose first term is $3$ and which sums to $2$ . If no such series exists, enter N/A in both spaces below: $2 = \sum _{n=0}^\infty \answer {3} \left ( \answer {- \frac {1}{2}} \right )^n$
Give an example of a geometric series whose first term is $3$ and which sums to $1$ . If no such series exists, enter N/A in both spaces below: $1 = \sum _{n=0}^\infty \answer {N/A} \left ( \answer {N/A} \right )^n$

Find the correct value of the constant in the blank below which makes the sum of the series equal to zero: $0 = \sum _{n=1}^\infty \frac {(-1)^{n-1}\answer {8} - 2^n}{3^n}$

One reasonable strategy is to separate the series into two geometric series and reindex to apply the standard formula.

Compute the sum of the infinite series below: $\sum _{n=1}^\infty \frac {1}{n(n+1)(n+2)} = \answer {\frac {1}{4}}.$

There is a closely-related telescoping series: $\frac {1}{n(n+1)} - \frac {1}{(n+1)(n+2)} = \frac {2}{n(n+1)(n+2)}$

Compute the sum of the infinite series below. $\sum _{n=1}^\infty \frac {2}{n(n+2)} = \answer {\frac {3}{2}}$

Expand the terms using partial fractions and compute several partial sums by hand. What you get is something like a telescoping series, but cancellations occur in a slightly different way than usual.

$\frac {2}{n(n+2)} = \frac {1}{n} - \frac {1}{n+2}.$

The formula for a general partial sum is $\frac {2}{1 \cdot 3} + \cdots + \frac {2}{N \cdot (N+2)} = 1 + \frac {1}{2} - \frac {1}{N+1} - \frac {1}{N+2}$

Use your answer to the previous exercise to compute the sum of the series: $\sum _{n=3}^\infty \frac {1}{n(n-2)} = \answer {\frac {3}{4}}$
Use the answer you just found to compute the sum of the series: $\sum _{n=4}^\infty \frac {1}{n(n-2)} = \answer {\frac {5}{12}}$
To drop the $n=3$ term, you can just calculate it and then subtract it from the previous series.

Sample Quiz Questions

Compute the exact value of the infinite series $\sum _{n=1}^\infty \ln \left ( \frac {1 + n^{-1}}{1 + (n+1)^{-1}} \right ).$

The series is not a geometric series or Taylor series, we compute the first few partial sums: $\begin {aligned} S_1 & = \ln \left ( \frac {1 + 1}{1 + 2^{-1}} \right ) = \ln \left ( \frac {2}{\frac {3}{2}} \right ) \\ S_2 & = \ln \left ( \frac {1 + 1}{1 + 2^{-1}} \right ) + \ln \left ( \frac {1 + 2^{-1}}{1 + 3^{-1}} \right ) = \ln \left ( \frac {1 + 1}{1 + 3^{-1}} \right ) = \ln \left ( \frac {2}{\frac {4}{3}} \right ) \\ S_3 & = \ln \left ( \frac {2}{1 + 3^{-1}} \right ) + \ln \left ( \frac {1 + 3^{-1}}{1 + 4^{-1}} \right ) = \ln \left ( \frac {2}{1 + 4^{-1}} \right ) = \ln \left ( \frac {2}{\frac {5}{4}} \right ) \\ & \ \vdots \\ S_n & = \ln \left ( \frac {2}{1 + (n+1)^{-1}} \right ). \end {aligned}$ In particular, writing the sum of logarithms as a logarithm of a product leads to substantial cancellation. By letting $n \rightarrow \infty$ , we get $S_n \rightarrow \ln 2$ .

Press...	...to do
left/right arrows	Move cursor
shift+left/right arrows	Select region
ctrl+a	Select all
ctrl+x/c/v	Cut/copy/paste
ctrl+z/y	Undo/redo
ctrl+left/right	Add entry to list or column to matrix
shift+ctrl+left/right	Add copy of current entry/column to to list/matrix
ctrl+up/down	Add row to matrix
shift+ctrl+up/down	Add copy of current row to matrix
ctrl+backspace	Delete current entry in list or column in matrix
ctrl+shift+backspace	Delete current row in matrix

Type...	...to get
norm	$\|\|\blue{[?]}\|\|$
text	$\text{\blue{[?]}}$
sym_name	$\backslash\texttt{\blue{[?]}}$
abs	$\left\|\blue{[?]}\right\|$
sqrt	$\sqrt{\blue{[?]}}$
paren	$\left(\blue{[?]}\right)$
floor	$\lfloor \blue{[?]} \rfloor$
factorial	$\blue{[?]}!$
exp	${\blue{[?]}}^{\blue{[?]}}$
sub	${\blue{[?]}}_{\blue{[?]}}$
frac	$\dfrac{\blue{[?]}}{\blue{[?]}}$
int	$\displaystyle\int{\blue{[?]}}d\blue{[?]}$
defi	$\displaystyle\int_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}d\blue{[?]}$
deriv	$\displaystyle\frac{d}{d\blue{[?]}}\blue{[?]}$
sum	$\displaystyle\sum_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
prod	$\displaystyle\prod_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
root	$\sqrt[\blue{[?]}]{\blue{[?]}}$
vec	$\left\langle \blue{[?]} \right\rangle$
mat	$\left(\begin{matrix} \blue{[?]} \end{matrix}\right)$
*	$\cdot$
infinity	$\infty$
arcsin	$\arcsin\left(\blue{[?]}\right)$
arccos	$\arccos\left(\blue{[?]}\right)$
arctan	$\arctan\left(\blue{[?]}\right)$
sin	$\sin\left(\blue{[?]}\right)$
cos	$\cos\left(\blue{[?]}\right)$
tan	$\tan\left(\blue{[?]}\right)$
sec	$\sec\left(\blue{[?]}\right)$
csc	$\csc\left(\blue{[?]}\right)$
cot	$\cot\left(\blue{[?]}\right)$
log	$\log\left(\blue{[?]}\right)$
ln	$\ln\left(\blue{[?]}\right)$
alpha	$\alpha$
beta	$\beta$
gamma	$\gamma$
delta	$\delta$
epsilon	$\epsilon$
zeta	$\zeta$
eta	$\eta$
theta	$\theta$
iota	$\iota$
kappa	$\kappa$
lambda	$\lambda$
mu	$\mu$
nu	$\nu$
xi	$\xi$
omicron	$\omicron$
pi	$\pi$
rho	$\rho$
sigma	$\sigma$
tau	$\tau$
upsilon	$\upsilon$
phi	$\phi$
chi	$\chi$
psi	$\psi$
omega	$\omega$
Gamma	$\Gamma$
Delta	$\Delta$
Theta	$\Theta$
Lambda	$\Lambda$
Xi	$\Xi$
Pi	$\Pi$
Sigma	$\Sigma$
Phi	$\Phi$
Psi	$\Psi$
Omega	$\Omega$

Sample Quiz Questions

Controls

Symbols

Settings