Question

Are the following statements true or false?Give reasons for your answers and explain in details.
(a)Every subsequence of the sequence {1\n^(2)} is convergent
(b)The function f(x)=x^2+|x|,is differentiable at x=-1
(c)Every infinite set is an open set.
(d)A necessary condition for a function f to be integrable is that it is continuous.
(e)The sum of the series 3n∑r=1 1/3n+2r as n→∞ can be calculated by evaluating the integral ∫ 1/3+2x dx from 0 to 3.

Answer 1

Question # 85540, Math / Real Analysis

Task: Are the following statements true or false? Give reasons for your answers and explain in details.

(a) Every subsequence of the sequence $\{1/n^2\}$ is convergent.

(b) The function $f(x) = x^2 + |x|$ is differentiable at $x = -1$.

(c) Every infinite set is an open set.

(d) A necessary condition for a function $f$ to be integrable is that it is continuous.

(e) The sum of the series $\sum_{r=1}^{3n}(1/3n + 2r)$ as $n \to \infty$ can be calculated by evaluating the integral $\int_0^3 (1/3 + 2x) \, dx$.

Solution:

(a) True. A sequence converges if and only if every subsequence converges. $\{1/n^2\}$ is convergent $\Rightarrow$ every subsequence of this sequence is convergent.

(b) True. For $x < 0$ we have $f(x) = x^2 - x$ and $f'(x) = 2x - 1$, $f'(-1) = -3$.

(c) False. The set $\{x: -1 \leqslant x \leqslant 1\} = [-1, 1]$ is infinite set but closed in $\mathbb{R}$.

(d) True. If $f$ is continuous then $f$ is Riemann integrable.

(e) False. $\sum_{r=1}^{3n}(1/3n + 2r) = 1 + 3n(3n + 1) \to \infty$ as $n \to \infty$. But $\int_0^3 (1/3 + 2x) \, dx = 10$.

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