Problem 2
The serien \(\sum_{n=0}(2 x)^{n}\) converges if
\((a)-1 \leq x \leq 1\)
\(\begin{array}{lll}\text { (b) }-\frac{1}{2}
Problem 3
The series \(\frac{2}{1^{2}}-\frac{3}{2^{2}}+\frac{4}{3^{2}}-\frac{5}{4^{2}}+\ldots\) in (a) conditionally convergent (b) nhsolutely convergent (c) divergent. (d) none of the above.
Problem 4
Which one of the following series is not convergent? (a) \(\frac{1}{2 \sqrt{2}}+\frac{1}{3 \sqrt{3}}+\frac{1}{4 \sqrt{4}}+\ldots=\) (b) \(1 \frac{1}{2}-1 \frac{1}{3}+1 \frac{1}{4}-1 \frac{1}{5}+\ldots \infty\) (c) \(\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\ldots=\) (d) \(\quad x+x^{2}+x^{3}+x^{4}+\ldots-\) where \(|x|<1\).
Problem 5
The sum of the alternating harmanic series \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}\), .... (a) zero (b) infinite (c) \(\log 2\) (d) not defined as the series is not convergent.
Problem 7
\(\sum\left(1+\frac{1}{n}\right)^{-n}\) is (c) convergent. (b) cseillatory (c) divergent.
Problem 8
\(1-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}-\frac{1}{\sqrt{4}}+\ldots\), is (a) oscillatory (b) conditionally convergent (c) divergent (d) absolutely convergent.
Problem 9
\(1+\frac{1}{2^{2}}-\frac{1}{3^{2}}-\frac{1}{4^{2}}+\frac{1}{5^{2}}+\frac{1}{6^{2}}-\frac{1}{7^{2}}-\frac{1}{8^{2}}+\ldots \cdots\) (a) conditionally convergent (b) convergent (c) oscillatory (d) divergent.
Problem 10
\(\int_{0}^{1}\left(\sum \frac{x^{n}}{n^{2}}\right) d x=\) (a) \(\sum_{n=0}^{\infty} \frac{1}{n(n+1)}\) (b) \(\sum_{n=1}^{\infty} \frac{1}{n^{2}(n-1)}\) (c) \(\sum_{n=0}^{\infty} \frac{1}{n(n-1)}\) (d) \(\sum_{n=1}^{-1} \frac{1}{n^{2}(n+1)}\).
Problem 23
The series \(\sqrt{\left(\frac{2^{n}-1}{3^{n}-1}\right)}\) is... (Convergent ete.)
Problem 25
In the neries \(\sum_{n=1}^{\infty} \frac{8 \tan ^{-1} n}{1+n^{2}}\) convergent?
