3.2 Series

Write out the first five partial sums corresponding to the series $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n}$.

You don't need to simplify the terms.

Every student who comes to class brings their instructor cookies, and leaves them on the instructor's desk. Let $C_k$ be the total number of cookies on the instructor's desk after the $k$th student comes.

If $C_{11}=20$, and $C_{10}=17$, how many cookies did the 11th student bring to class?

Suppose the sequence of partial sums of the series $\displaystyle\sum_{n=1}^\infty a_n$ is $\{S_N\} = \left\{\dfrac{N}{N+1}\right\}$.

Suppose the sequence of partial sums of the series $\displaystyle\sum_{n=1}^\infty a_n$ is $\{S_N\} = \left\{(-1)^N+\dfrac{1}{N}\right\}$.

What is $\{a_n\}$?

Let $f(N)$ be a formula for the $N$th partial sum of $\displaystyle\sum_{n=1}^\infty a_n$. (That is, $f(N)=S_N$.) If $f'(N)<0$ for all $N > 1$, what does that say about $a_n$?

Suppose the triangle outlined in red in the picture below has area one.

Suppose the square outlined in red in the picture below has area one.

In the style of Questions 6 and 7, draw a picture that represents $\displaystyle\sum_{n=1}^{\infty}\frac{1}{3^n}$ as an area.

Evaluate $\displaystyle\sum_{n=0}^{100}\frac{1}{5^n}$.

Every student who comes to class brings their instructor cookies, and leaves them on the instructor's desk. Let $C_k$ be the total number of cookies on the instructor's desk after the $k$th student comes.

If $C_{20}=53$, and $C_{10}=17$, what does $C_{20}-C_{10}=36$ represent?

Evaluate $\displaystyle\sum_{n=50}^{100}\frac{1}{5^n}$. (Note the starting index.)

Suppose $\displaystyle\sum_{n=1}^\infty a_n = A$, $\displaystyle\sum_{n=1}^\infty b_n = B$, and $\displaystyle\sum_{n=1}^\infty c_n = C$.

Evaluate $\displaystyle\sum_{n=1}^\infty\left( a_n+b_n+c_{n+1}\right)$.

Suppose $\displaystyle\sum_{n=1}^\infty a_n = A$, $\displaystyle\sum_{n=1}^\infty b_n = B \neq 0$, and $\displaystyle\sum_{n=1}^\infty c_n = C$.

True or false: $\displaystyle\sum_{n=1}^\infty\left(\dfrac{ a_n}{b_n}+c_{n}\right) = \frac{A}{B}+C$.

To what value does the series $\displaystyle 1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27}+ \frac{1}{81}+ \frac{1}{243} + \cdots$ converge?

Evaluate $\displaystyle \sum_{k=7}^{\infty} \frac{1}{8^k}$

Show that the series $\displaystyle \sum_{k=1}^{\infty} \bigg( \frac{6}{k^2} - \frac{6}{(k+1)^2} \bigg)$ converges and find its limit.

Find the sum of the convergent series $\displaystyle\sum\limits_{n=3}^\infty \bigg( \!\cos\Big( \frac \pi n \Big) - \cos\Big( \frac \pi{n+1} \Big) \bigg)$.

The $n^{\rm th}$ partial sum of a series $\displaystyle \sum_{n=1}^{\infty}a_n$ is known to have the formula $\displaystyle s_n = \frac{1+3n}{5+4n}$.

Find the sum of the series $\displaystyle\sum\limits_{n=2}^\infty \frac{3\cdot 4^{n+1}}{8\cdot 5^n}$. Simplify your answer completely.

Relate the number $0.2\bar{3} = 0.233333\ldots$ to the sum of a geometric series, and use that to represent it as a rational number (a fraction or combination of fractions, with no decimals).

Express $2.656565\ldots$ as a rational number, i.e. in the form $p/q$ where $p$ and $q$ are integers.

Express the decimal $0.\overline{321}=0.321321321\ldots$ as a fraction.

Find the value of the convergent series

Simplify your answer completely.

Evaluate

Find the sum of the series $\displaystyle\sum_{n=0}^\infty\frac{1+3^{n+1}}{4^n}$.

Evaluate $\displaystyle\sum_{n=5}^\infty \log\left(\frac{n-3}{n}\right)$.

Evaluate $\displaystyle\sum_{n=2}^\infty \left(\frac{2}{n}-\frac{1}{n+1}-\frac{1}{n-1} \right)$.

An infinitely long, flat cliff has stones hanging off it, attached to thin wire of negligible mass. Starting at position $x=1$, every metre (at position $x$, where $x$ is some whole number) the stone has mass $\dfrac{1}{4^x}$ kg and is hanging $2^x$ metres below the top of the cliff.

Find the combined volume of an infinite collection of spheres, where for each whole number $n=1,\,2,\,3,\,\ldots$ there is exactly one sphere of radius $\dfrac{1}{\pi^n}$.

Evaluate $\displaystyle\sum_{n=3}^\infty \left(\frac{\sin^2 n}{2^n}+\frac{\cos^2(n+1)}{2^{n+1}} \right)$.

Suppose a series $\displaystyle\sum_{n=1}^\infty a_n$ has sequence of partial sums $\{S_N\}$, and the series $\displaystyle\sum_{N=1}^\infty S_N$ has sequence of partial sums $\{\mathscr S_M \}=\left\{ \displaystyle\sum_{N=1}^M S_N\right\}$.

If $\mathscr S_M = \dfrac{M+1}{M}$, what is $a_n$?

Create a bullseye using the following method:

Starting with a red circle of area 1, divide the radius into thirds, creating two rings and a circle. Colour the middle ring blue.

Continue the pattern with the inside circle: divide its radius into thirds, and colour the middle ring blue.

Continue in this way indefinitely: dividing the radius of the innermost circle into thirds, creating two rings and another circle, and colouring the middle ring blue.

What is the area of the red portion?