Practice Questions for NET JRF Real Analysis Assignment: Series of Real Numbers III

21. \(\sum_{n=1}^{\infty} \frac{1}{\left(n^{2}+1\right)^{k}}\) is convergent for

(a.) \(k<\frac{1}{2}\)
(b.) \(k>\frac{1}{2}\)
(c.) \(k=\frac{1}{2}\)
(d.) All values of \(k\)

22. \(\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{1}{3 n+k}\) is

(a.) \(\log \frac{4}{3}\)
(b.) \(\log \frac{3}{4}\)
(c.) \(\log \frac{3}{2}\)
(d.) \(\log \frac{5}{4}\)

23. \(\sum \frac{n^{n}}{(n+a)^{n}} a: a>0\)

(a.) Converges for all \(a>1\)
(b.) Converges for \(a>0\)
(c.) Converges for \(a \geq 0\)
(d.) Divergent for all \(a\)

24. Which of the following series are convergent?

(a.) \(\sum_{n=1}^{\infty} \frac{\sqrt{2 n^{2}+3}}{5 n^{3}+7}\)
(b.) \(\sum_{n=1}^{\infty} \frac{(n+1)^{n}}{n^{n+\frac{3}{2}}}\)
(c.) \(\sum_{n=1}^{\infty} \frac{1}{n} \sin \left(\frac{1}{n}\right)\)
(d.) None of these

25. Consider the sequence of rational numbers \(\left\{q_{k}\right\}_{k \geq 1}\) where \(q_{k}=\sum_{n=1}^{k} \frac{1}{10^{n^{2}}}\), i.e., the sequence is \(q_{1}=.1, q_{2}=.1001, q_{3}=.100100001\) etc.

Which of the following is true?

(a.) This sequence is bounded and convergent in \(\mathbb{Q}\)
(b.) This sequence is not bounded
(c.) This sequence is bounded, but not a Cauchy sequence
(d.) This sequence is bounded and Cauchy, but not convergent in \(\mathbb{Q}\)

26. Let \(\left\{a_{0}, a_{1}, a_{2}, \ldots\right\}\) be a sequence of real numbers. For any \(k \geq 1\), let \(s_{n}=\sum_{k=0}^{n} a_{2 k}\). Which of the following statements are correct?

(a.) If \(\lim _{n \rightarrow \infty} s_{n}\) exists, then \(\sum_{m=0}^{\infty} a_{m}\) exists.
(b.) If \(\lim _{n \rightarrow \infty} s_{n}\) exists, then \(\sum_{m=0}^{\infty} a_{m}\) need not exist.
(c.) If \(\sum_{m=0}^{\infty} a_{m}\) exists, then \(\lim _{n \rightarrow \infty} s_{n}\) exists.
(d.) If \(\sum_{m=0}^{\infty} a_{m}\) exists, then \(\lim _{n \rightarrow \infty} s_{n}\) need not exist.

27. If \(\sum_{n=1}^{\infty} a_{n}\) is absolutely convergent, then which of the following is NOT true?

(a.) \(\sum_{m=n}^{\infty} a_{m} \rightarrow 0\) as \(n \rightarrow \infty\).
(b.) \(\sum_{n=1}^{\infty} a_{n} \sin n\) is convergent.
(c.) \(\sum_{n=1}^{\infty} e^{a_{n}}\) is divergent.
(d.) \(\sum_{n=1}^{\infty} a_{n}^{2}\) is divergent.

28. Using the fact that \(\sum_{1}^{\infty} \frac{1}{n^{2}}=\frac{\pi^{2}}{6}\), \(\sum_{1}^{\infty} \frac{1}{(2 n+1)^{2}}\) equals

(a.) \(\frac{\pi^{2}}{12}\)
(b.) \(\frac{\pi^{2}}{12}-1\)
(c.) \(\frac{\pi^{2}}{8}\)
(d.) \(\frac{\pi^{2}}{8}-1\)

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29. Pick out the series which are absolutely convergent:

(a.) \(\sum_{n=1}^{\infty}(-1)^{n} \frac{\cos n \alpha}{n^{2}}\) where \(\alpha \in \mathbb{R}\) is a fixed real number.
(b.) \(\sum_{n=1}^{\infty}(-1)^{n} \frac{n \log n}{e^{n}}\)
(c.) \(\sum_{n=1}^{\infty}(-1)^{n} \frac{n}{n+2}\)
(d.) None of these

30. If \(\sum a_{n}\) is a series of positive and negative terms, and \(\sum q_{n}\) and \(\sum p_{n}\) are series of negative and positive terms, respectively, and if \(\sum a_{n}\) is conditionally convergent, then

(a.) \(\sum p_{n}\) is convergent but \(\sum q_{n}\) is not.
(b.) \(\sum p_{n}\) is divergent but \(\sum q_{n}\) is not.
(c.) Both \(\sum p_{n}\) and \(\sum q_{n}\) are convergent.
(d.) Both \(\sum p_{n}\) and \(\sum q_{n}\) are divergent.