Question

(a) If a seriesE1an converges, then lim,n-0 an = 0. m=1 (b) If f O(g), then f(x) < g(x) for all sufficiently large . R is any one-to-one differentiable function, then f-1 is (c) If f: R differentiable on R (d) The sequence a1, a2, a3, -.. defined by max{ sin 1, sin 2,-.- , sin n} an converges (e) If a power series 0«na" converges absolutely at x also converge absolutely at x = -2 2, then it must

Please let me know whether true or false

If false, please give me the counter example!

Answer 1

a) A necessary condition for
Σ n=1
to converge is
lim an0 n-too
. Hence the statement is true.

b)
f = O(g)
if and only if there exists a positive real number and a real number $x_0$ such that
F(x Mg(a)
for all .

Hence the given question is true for and arbitrary $x_0$

c) We have, if
R f : R
is one-one differentiable and with inverse function $f^{-1}$ and
(a)0
, then the inverse function is differentiable at a and
f1Y(a) = (f1 (a)

Hence the option is false.

d) $a_n = max \{ \sin 1, \sin 2, ..., \sin n \}$

Clearly $a_n$ is bounded, since $|\sin n|\leq 1$ .

But $\sin$ value oscillates between -1 and 1, but maximum value is 1, obtained at n=90

Hence

lim an1 n-too

Hence it is true

e) If the power series $\sum_{n=1}^{\infty}a_nx^n$ converges absolutely at , then the radius of convergence is 2. Hence it also converges absolutely at .

Hence it is true.