Gerard Sayson TMUA Mock Paper 1 · Question 17

The following theorem is quoted during a lecture about mathematical analysis:

Weierstrass $M$ -test

Let $f_1,f_2,\ldots$ be a sequence of functions $f_i:X\to\mathbb R$ . If there are constants $M_k$ for all $k\ge1$ such that $|f_k(x)|\le M_k$ for all $x\in X$ and $\sum_{k=1}^\infty M_k$ converges, then the series $\sum_{k=1}^\infty f_k(x)$ converges absolutely and uniformly on $X$ .

(It is not required to know the concepts of absolute and uniform convergence in this question.)

What is the negation of the above theorem?

AThere do not exist constants

M_k

for any

k\ge1

such that

|f_k(x)|\le M_k

for any

x\in X

and

\sum_{k=1}^\infty M_k

converges.

BThere exist constants

M_k

for any

k\ge1

such that

|f_k(x)|\le M_k

for any

x\in X

\sum_{k=1}^\infty M_k

converges, and

\sum_{k=1}^\infty f_k(x)

does not converge absolutely or uniformly on

X

CThere exists a constant

M_k

for some

k\ge1

such that

|f_k(x)|\le M_k

for any

x\in X

\sum_{k=1}^\infty M_k

converges, and

\sum_{k=1}^\infty f_k(x)

does not converge absolutely and uniformly on

X

DThere exist constants

M_k

for any

k\ge1

such that

|f_k(x)|\le M_k

for any

x\in X

, and

\sum_{k=1}^\infty M_k

converges, and

\sum_{k=1}^\infty f_k(x)

does not converge absolutely and uniformly on

X

examday

Gerard Sayson TMUA Mock Paper 1 · Q17