Chapter 9.4, Problem 67E

### Calculus: Early Transcendental Fun...

6th Edition
Ron Larson + 1 other
ISBN: 9781285774770

Chapter
Section

### Calculus: Early Transcendental Fun...

6th Edition
Ron Larson + 1 other
ISBN: 9781285774770
Textbook Problem

# PUTNAM EXAM CHALLENGEProve that if ∑ n = 1 ∞ a n is a convergent scries of positive real numbers, then so is ∑ n = 1 ∞ ( a n ) n / ( n + 1 )

To determine

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

Explanation

Given:

The series n=1an is Convergent series of real positive number.

Formula Used:

Direct comparison test.

Let 0<anbn for all n.

If n=1an, and n=1bn are convergent series, then n=1can and n=1(an+bn) and

1) n=1can=cn=1an

2) n=1(an+bn)=n=1an+n=1bn.

Proof:

Consider n=1an is convergent series of positive real numbers.

Consider two cases:

Case 1:

If an12(n+1) then,

(an)1/(n+1)(12(n+1))1/(n+1)(an)1/(n+1)12

And,

(an)nn+1=(an)(an)(n+1)=2an

Case 2:

If an12(n+1) then,

(an)n/(n+1)(12(n+1))n/(n+1)12n

Compare both case 1 and case 2.

(an)nn+12an+12n

Use direct comparison test

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