Chapter 1: Q7P (page 1)

For the following series, write formulas for the sequences $a_{n}, S_{n}, and R_{n}$ , and find the limits of the sequences as $n \to \infty$ (if the limits exist).
$\frac{3}{1 \cdot 2} - \frac{5}{2 \cdot 3} + \frac{7}{3 \cdot 4} - \frac{9}{4 \cdot 5} + . . . . .$

Short Answer

Expert verified

Hence, the required formulae is:

$\lim_{k \to \infty} a_{k} = 0 \lim_{k \to \infty} S_{k} = 1 \lim_{k \to \infty} R_{k} = 0$

Step by step solution

Sum of the Series

The sum of the k terms of the series which is in geometric progression, with first term a and common ratio r, is given by:

$S_{k} = \frac{a (1 - r^{k})}{1 - r}$

Find the General Term

The given series is: $\frac{3}{1 \cdot 2} - \frac{5}{2 \cdot 3} + \frac{7}{3 \cdot 4} - . . . . . .$

The series can be represented as:

$\frac{3}{1 \cdot 2} - \frac{5}{2 \cdot 3} + \frac{7}{3 \cdot 4} - . . . . . . = \sum_{k = 1}^{\infty} {(- 1)}^{k + 1} \frac{2 k + 1}{k^{2} + k}$

Clearly, the general term will be:

$\begin{array}{rcl} a_{k} & = & {(- 1)}^{k + 1} \frac{2 k + 1}{k^{2} + k} \\ = & {(- 1)}^{k + 1} \frac{2 k + 1}{k (k + 1)} \end{array}$

Using partial fraction method, we have:

$\begin{array}{rcl} \frac{2 k + 1}{k (k + 1)} & = & \frac{A}{k} + \frac{B}{k + 1} \\ 2 k + 1 & = & (A + B) k + A \\ A & = & 1 \\ B & = & 1 \end{array}$

So, we get:

$\begin{array}{rcl} a_{k} & = & {(- 1)}^{k + 1} \frac{2 k + 1}{k (k + 1)} \\ = & {(- 1)}^{k + 1} \{\frac{1}{k} + \frac{1}{k + 1}\} \\ \lim_{k \to \infty} a_{k} & = & 0 \end{array}$

Find the Sum

Now, let us substitute $k = 1, 2, 3, . . . . . . . . . ., (n - 1), n$ , we get:

$\begin{array}{rcl} S_{k} & = & \frac{1}{1} + \frac{1}{2} - \frac{1}{2} - \frac{1}{3} + \frac{1}{3} + \frac{1}{4} - . . . . . . . + {(- 1)}^{k} \{\frac{1}{k - 1} + \frac{1}{k}\} + {(- 1)}^{k} \{\frac{1}{k} + \frac{1}{k + 1}\} \\ = & 1 + \frac{{(- 1)}^{k + 1}}{k + 1} \end{array}$

Find the Sequences

Now, we have:

$\begin{array}{rcl} S & = & \lim_{k \to \infty} S_{k} \\ = & \lim_{k \to \infty} [1 + \frac{{(- 1)}^{k + 1}}{k + 1}] \\ = & 1 \end{array}$

The sequence can be calculated as follows:

$\begin{array}{rcl} R_{k} & = & S - S_{k} \\ = & 1 - 1 - \frac{{(- 1)}^{k + 1}}{k + 1} \\ = & \frac{- {(- 1)}^{k + 1}}{k + 1} \\ \lim_{k \to \infty} R_{k} & = & 0 \end{array}$

Hence, this is the required answer.

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For the following series, write formulas for the sequences $a_{n}, S_{n}, and R_{n}$ , and find the limits of the sequences as $n \to \infty$ (if the limits exist).
$\frac{3}{1 \cdot 2} - \frac{5}{2 \cdot 3} + \frac{7}{3 \cdot 4} - \frac{9}{4 \cdot 5} + . . . . .$

Short Answer

Step by step solution

Sum of the Series

Find the General Term

Find the Sum

Find the Sequences

One App. One Place for Learning.

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For the following series, write formulas for the sequences an,Sn,andRn, and find the limits of the sequences as n→∞ (if the limits exist). 31·2-52·3+73·4-94·5+.....

Short Answer

Step by step solution

Sum of the Series

Find the General Term

Find the Sum

Find the Sequences

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Thermodynamics

Atoms and Radioactivity

Fluids

Further Mechanics and Thermal Physics

Linear Momentum

Measurements

Study anywhere. Anytime. Across all devices.

For the following series, write formulas for the sequences $a_{n}, S_{n}, and R_{n}$ , and find the limits of the sequences as $n \to \infty$ (if the limits exist).
$\frac{3}{1 \cdot 2} - \frac{5}{2 \cdot 3} + \frac{7}{3 \cdot 4} - \frac{9}{4 \cdot 5} + . . . . .$