Chapter 9.4, Problem 50E

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Chapter
Section

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

# Using a Series Consider the series ∑ n = 1 ∞ 1 ( n + 2 ) 2 , (a) Verify that the series converges.(b) Use a graphing utility to complete the table. n 5 10 20 50 100 S n (c) The sum of the series is ( π 2 / 6 ) − ( 5 / 4 ) . Find the sum of the series ∑ n = 6 ∞ 1 ( n + 2 ) 2 , (d) Use a graphing utility to find the sum of the series ∑ n = 15 ∞ 1 ( n + 2 ) 2 ,

(a)

To determine

To prove: The series n=1(1(n+2)2) is convergent series.

Explanation

Given:

The series is n=1(1(n+2)2).

Formula used:

If Un>0 and Vn>0 are the nth-term of two series of infinite terms and, UnVn.

then as per the Direct comparison test, if Vn converges, then Un also converges.

The series of the form V=n=11np is known as p-series which is convergent if p>1 and divergent if 0<p1.

Proof:

The series is,

n=1(1(n+2)2)

Let, n=1(1(n+2)2) be an infinite series of positive termswith nth term denoted by Un=1(n+2)2 and V=n=11n2 be another infinite series of positive terms with nth-term denoted by Vn=1(n)2.

It is known that,

(n+2)>n

Square both the sides of the above equation,

(n+2)2>n21(n+2)2<1n2

Take summation both the sides,

n=1</

(b)

To determine

To calculate: The partial sum of the series, n=1(1(n+2)2) for n terms given in the table with the help of graphical utility.

(c)

To determine

To calculate: The sum of the series n=161(n+2)2 when the sum of the series, n=1(1(n+2)2) is π2654.

(d)

To determine

To calculate: The sum of the series, n=151(n+2)2 with the help of a graphical calculator.

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