Chapter 7: Q. 35 (page 615)

Provide the first five terms of the sequence of partial sums for the given series.
$\sum_{n = 0}^{\infty} \frac{(- 1)^{n}}{(2 n)!}$

Short Answer

Expert verified

The first five terms of the partial sums for the series $\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(2 n)!}$ are, $\{1, \frac{1}{2}, \frac{13}{24}, \frac{389}{720}, \frac{21785}{40320}\}$ .

Step by step solution

Step 1. Given Information

$\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(2 n)!}$ .

Step 2. Find the First term

Substitute $n = 0$ in the given series.

\begin{array}{l} \frac{(- 1)^{n}}{(2 n)!} = \frac{(- 1)^{0}}{(2 \times 0)!} \\ = \frac{1}{0!} \\ = 1 \end{array}

The first term is, $1 .$

Step 3. Find the Second term

Substitute $n = 1$ in the given series.

\begin{array}{l} \frac{(- 1)^{n}}{(2 n)!} = \frac{(- 1)^{1}}{(2 \times 1)!} \\ = \frac{- 1}{2!} \\ = - \frac{1}{2} \end{array}

The second term is, $- \frac{1}{2}$ .

Step 4. Find the third term.

Substitute $n = 2$ in the given series.

$\frac{{(- 1)}^{n}}{(2 n)!} = \frac{{(- 1)}^{2}}{(2 (2))!} = \frac{(1)}{(4)!} = \frac{1}{24}$

The third term is, $\frac{1}{24}$ .

Step 5. Find the fourth term.

Substitute $n = 3$ in the given series.

$\frac{{(- 1)}^{n}}{(2 n)!} = \frac{{(- 1)}^{3}}{(2 (3))!} = - \frac{1}{720}$

The fourth term is, $- \frac{1}{720}$ .

Step 6. Find the fifth term.

Substitute $n = 4$ in the given series.

$\frac{{(- 1)}^{n}}{(2 n)!} = \frac{{(- 1)}^{4}}{(2 (4))!} = \frac{1}{40, 320}$

The fifth term is, $\frac{1}{40, 320}$ .

Step 7. The first five terms in the sequence of partial sums are,

$S_{1} = 1 S_{2} = S_{1} + a_{2} = 1 - \frac{1}{2} = \frac{1}{2} S_{3} = S_{2} + a_{3} = \frac{1}{2} + \frac{1}{24} = \frac{13}{24} S_{4} = S_{3} + a_{4} = \frac{13}{24} - \frac{1}{720} = \frac{389}{720} S_{5} = S_{4} + a_{5} = \frac{389}{720} + \frac{1}{40, 320} = \frac{21785}{40, 320}$

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Provide the first five terms of the sequence of partial sums for the given series.
$\sum_{n = 0}^{\infty} \frac{(- 1)^{n}}{(2 n)!}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Find the First term

Step 3. Find the Second term

Step 4. Find the third term.

Step 5. Find the fourth term.

Step 6. Find the fifth term.

Step 7. The first five terms in the sequence of partial sums are,

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Provide the first five terms of the sequence of partial sums for the given series.∑n=0∞ (−1)n(2n)!

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Find the First term

Step 3. Find the Second term

Step 4. Find the third term.

Step 5. Find the fourth term.

Step 6. Find the fifth term.

Step 7. The first five terms in the sequence of partial sums are,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Theoretical and Mathematical Physics

Mechanics Maths

Applied Mathematics

Geometry

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Provide the first five terms of the sequence of partial sums for the given series.
$\sum_{n = 0}^{\infty} \frac{(- 1)^{n}}{(2 n)!}$