Sequences and Series

Arithmetic Sequences/Progressions

If the first term of an arithmetic sequence is 3 and the common difference is 2, find the sum of the first 10 terms.

Geometric Sequences/Progressions

Find the sum of the geometric sequence

3, 6, 12, 24, \dots

up to the nth term.

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Finding the Next Term of the Sequence

Consider the sequence

a_{n}

where

a_{1} = 1

a_{2} = 2

, and for

n \geq 3

a_{n} = 2 a_{n - 1} - a_{n - 2}

. Find the next term

a_{6}

of the sequence?

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Finding the nth Term Given a List of Numbers

Given a sequence of numbers: 2, 5, 10, 17, find the formula for the nth term and calculate the 8th term of the sequence.

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Finding the nth Term

Find the 10th term of the sequence defined by

a_{n} = 2 n^{2} + 3 n + 1

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Finding the Sum of First n Terms

Find the sum of the first 100 terms of the sequence defined by

a_{n} = n^{2}

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Expanding Series Notation

Question 10 ( 4 points) Let

g (x) = x \cos (3 x)

. Find the Maclaurin series of its derivative,

g^{'} (x)

None of the other answers are correct.

3 x - \frac{9}{2} x^{3} + \frac{81}{40} x^{5} + \dots

3 - \frac{3}{2} x^{2} + \frac{5}{8} x^{4} + \dots

1 - \frac{27}{2} x^{2} + \frac{135}{8} x^{4} + \dots

3 - \frac{27}{2} x^{2} + \frac{81}{8} x^{4} + \dots

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Finding the Sum of the Series

Find the indicated sum.

\sum_{S_{7} = 1}^{7} 5^{k}

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Finding the Sum of the Infinite Geometric Series

Find the sum of

\sum_{n = 0}^{\infty} \frac{n (n + 1)}{3^{n - 1}}

by identifying it as the value of the derivative of the series

g (x) = \sum_{n = 0}^{\infty} (n + 1) x^{n} = \frac{1}{(1 - x)^{2}} for | x | < 1 .

a. Take the derivative of the series given by

g (x)

. Write the derivative of the series in the first box and the derivative of the rational expression in the second box.

g^{'} (x) = \sum_{n = 1}^{\infty} ◻ =

b. Find the value of

g^{'} (\frac{1}{3}) = \sum_{n = 0}^{\infty} \frac{n (n + 1)}{3^{n - 1}}

g^{'} (\frac{1}{3}) =

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Determining if a Series is Divergent

For the sequence, determine if the divergence test applies and either state that

lim_{n \to \infty} a_{n}

does not exist or find

lim_{n \to \infty} a_{n}

. (If an answer does not exist, enter DNE.)

\begin{array}{r} a_{n} = \frac{2^{n} + 3^{n}}{10^{n / 2}} \\ lim_{n \to \infty} a_{n} = \end{array}

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Using the Integral Test for Convergence

Test the series below for convergence using the Ratio Test.

\sum_{n = 1}^{\infty} \frac{n^{4}}{{1.5}^{n}}

The limit of the ratio test simplifies to

lim_{n \to \infty} | f (n) |

where

f (n) =

The limit is: (enter oo for infinity if needed) Based on this, the series Select an answer

✓

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Determining if an Infinite Series is Convergent Using Cauchy's Root Test

Find the series' radius of convergence.

\sum_{n = 1}^{\infty} \frac{(x - 1)^{n}}{\ln (n + 1)}

A) 1 B)

\infty

, formall

x

C) 0 D) 2

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Arithmetic Sequences/Progressions

Geometric Sequences/Progressions

Finding the Next Term of the Sequence

Finding the nth Term Given a List of Numbers

Finding the nth Term

Finding the Sum of First n Terms

Expanding Series Notation

Finding the Sum of the Series

Finding the Sum of the Infinite Geometric Series

Determining if a Series is Divergent

Using the Integral Test for Convergence

Determining if an Infinite Series is Convergent Using Cauchy's Root Test

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