Department of Mathematical and Computational Sciences
National Institute of Technology Karnataka, Surathkal
Engineering Mathematics II - MA111 (2023-24)
Problem Sheet 5
Taylor Series and Maclaurin Series
1. What is the radius of convergence for the Maclaurin’s series of
(a) tan−1x(b) log(1 + x)(c) 1
(1 + x)2
2. Find the Taylor’s Polynomial of order 5 for the following functions.
(a) tan xabout x= 0
(b) tan−1xabout x= 0
(c) sin xabout x=π/3
(d) ex2−2xabout x= 1
3. Find the Maclaurin’s series for the following functions.
(a) f(x) = 1
1 + x2(b) f(x) = ecos x
4. Find the Taylor’s series of the following functions. Also, find the radius of conver-
gence and interval of convergence in each case.
(a) 1
1−xabout x=−1 (b) 1
2 + x2about x= 2 (c) 1
xabout x= 10
5. Using Taylor’s series for cos xprove that 1
2−x2
24 <1−cos x
x2<1
2, x = 0.
6. Show that the Taylor’s series generated by f(x) = 1
1−xat x= 0 converges to f(x)
for x∈(−1/2,1/2).
7. Approximation property of Taylor’s polynomial: Suppose that f(x) is differentiable
on an interval centered at x=aand that
g(x) = b0+b1(x−a) + · ·· +bn(x−a)n
is a polynomial of degree nwith constant coefficients b0, b1. . . bnand let E(x) =
f(x)−g(x).
Show that if we impose on gthe conditions (i) E(a) = 0 and (ii) lim
x→a
E(x)
(x−a)n= 0,
then
g(x) = f(a) + f′(a)(x−a) + (x−a)2
2! f′′(a) + · · · +(x−a)n
n!f(n)(a).
8. Show that a function f(x) defined by a power series as f(x) =
∞
X
n=0
anxnwith a
positive radius of convergence Rhas a Taylor’s series that converges to f(x), for all
x∈(−R, R).
