Indian Institute of Technology Guwahati

Department of Mathematics

MA 101 Mathematics-I Tutorial Sheet-3

Date: 12-Oct-2012

Topics Covered:

Series, Tests for convergence, Rearrangements of terms Power Series, Radius of Convergence

1. Check whether the following series are convergent, absolutely convergent or divergent.

(a)

1

1+anwhere a > 0.

(b)

n!nn

(c)

n2

1(logn)n

(d)

(n(n+1))2n!

2. (a) Let (xn) be a sequence of real numbers. The telescoping seriesn=1

(xn xn+1) isconvergent if and only if the sequence (xn) is convergent. Find out the sum in this case.

(b) (Cauchy's Condensation Test:) Let (xn) be a decreasing sequence of non-negativereal numbers. The the series

xn converges if and only if

n0

2nx2n converges.

(c) Using Cauchy's Condensation Test, (or otherwise?) discuss and prove the convergence

of p-series

1npfor a xed real number p.

3. Let (xn) be such that limn

n2xn exists. Then show that

xn converges absolutely.

4. Let (xn) be a monotonically decreasing sequence of positive real numbers. If limn

xn+1 = 0

then show that the alternating series

n1

(1)n+1xn is convergent.

Can you drop the condition that (xn) is monotonically decreasing?

5. Show that the series

(1)n+1nis conditionally convergent.

6. Complete the proof of Root Test for checking convergence of series.

7. Determine the radius of convergence and sum of the power series

n1

anXnwhere an is given

by

(a) a1 = 0 and an =1

lognfor n 2(b)

np

n!for a xed p > 0

(c)

(1)n+1n

(d) 1 + x+ x2

2!+ x

3

3!+