Chapter 12: Q. 59 (page 1174)

In the following exercise, expand the partial sum and find its value
$\sum_{n = 1}^{5} \frac{n}{n + 1}$

Short Answer

Expert verified

Hence,

$\sum_{n = 1}^{5} \frac{n}{n + 1} = \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \frac{5}{6} = \frac{71}{20}$

Step by step solution

Step 1. Given information

$\begin{array}{l} Thepartialsumisgivenas \\ \sum_{n = 1}^{5} \frac{n}{n + 1} \\ Wehavetoexpandandfindthevalueofthepartialsum. \end{array}$

Step 2. Expanding the partial sum

$\begin{array}{l} Wesubstitutethevalues 1, 2, 3, 4, 5inorder \\ \Rightarrow \sum_{n = 1}^{5} \frac{n}{n + 1} = \frac{1}{1 + 1} + \frac{2}{2 + 1} + \frac{3}{3 + 1} + \frac{4}{4 + 1} + \frac{5}{5 + 1} \\ = \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \frac{5}{6} \end{array}$

Step 3. Calculating the sum

Now,

$\begin{array}{l} \Rightarrow \sum_{n = 1}^{5} \frac{n}{n + 1} = \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \frac{5}{6} = \frac{81}{20} \\ = \frac{1 \times 30 + 2 \times 20 + 3 \times 15 + 4 \times 12 + 5 \times 10}{60} \\ = \frac{30 + 40 + 45 + 48 + 50}{60} \\ = \frac{213}{60} \\ = \frac{71}{20} \end{array}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

In the following exercise, expand the partial sum and find its value
$\sum_{n = 1}^{5} \frac{n}{n + 1}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Expanding the partial sum

Step 3. Calculating the sum

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Theoretical and Mathematical Physics

Applied Mathematics

Logic and Functions

Mechanics Maths

Pure Maths

Study anywhere. Anytime. Across all devices.

In the following exercise, expand the partial sum and find its value ∑n=15nn+1

Short Answer

Step by step solution

Step 1. Given information

Step 2. Expanding the partial sum

Step 3. Calculating the sum

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Theoretical and Mathematical Physics

Applied Mathematics

Logic and Functions

Mechanics Maths

Pure Maths

Study anywhere. Anytime. Across all devices.

In the following exercise, expand the partial sum and find its value
$\sum_{n = 1}^{5} \frac{n}{n + 1}$