List of top Mathematics Questions on Series

Find the missing number in the series: \(2,\,6,\,12,\,20,\,30,\,?\)

Evaluate: \( \cot^{-1}(2) - \cot^{-1}(8) - \cot^{-1}(18) - \dots \)

Let \[ S=\frac{1}{2!5!}+\frac{1}{3!2!3!}+\frac{1}{5!2!1!}+\cdots \text{ up to 13 terms}. \] If $13S=\dfrac{2^k}{n!}$, $k\in\mathbb{N}$, then $n+k$ is equal to

$\cot^{-1}(2 \cdot 1^2) + \cot^{-1}(2 \cdot 2^2) + \cot^{-1}(2 \cdot 3^2) + \dots \dots \infty =$

$\cot^{-1}(2 \cdot 1^2) + \cot^{-1}(2 \cdot 2^2) + \cot^{-1}(2 \cdot 3^2) + \dots \dots \infty =$

$\cot^{-1}(2 \cdot 1^2) + \cot^{-1}(2 \cdot 2^2) + \cot^{-1}(2 \cdot 3^2) + \dots \dots \infty =$

Given the series
$\frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \cdots = \frac{\pi^4}{90},$
$\frac{1}{1^4} + \frac{1}{3^4} + \frac{1}{5^4} + \cdots = \alpha,$
$\frac{1}{2^4} + \frac{1}{4^4} + \frac{1}{6^4} + \cdots = \beta.$
Then find $\frac{\alpha}{\beta}$

If \[1 + \frac{\sqrt{3} - \sqrt{2}}{2\sqrt{3}} + \frac{5 - 2\sqrt{6}}{18} + \frac{9\sqrt{3} - 11\sqrt{2}}{36\sqrt{3}} + \frac{49 - 20\sqrt{6}}{180} + \cdots\] up to \(\infty = 2 \left( \sqrt{\frac{b}{a}} + 1 \right) \log_e \left( \frac{a}{b} \right)\), where \(a\) and \(b\) are integers with \(\gcd(a, b) = 1\), then (11a + 18b\) is equal to _________.

The coefficient of \(x^n\) in the expansion of \[\frac{e^{7x} + e^x}{e^{3x}}\] is:

The sum of the infinite series \(1 + \frac{5}{6} + \frac{12}{6^2} + \frac{22}{6^3} + \frac{35}{6^4} + \dots\) is equal to:

If \( A = 1 + r^a + r^{2a} + r^{3a} + \dots \infty \) and \( B = 1 + r^b + r^{2b} + r^{3b} + \dots \infty \), then \( \frac{a}{b} \) is equal.

There are four numbers of which the first three are in GP and the last three are in AP, whose common difference is 6. If the first and the last numbers are equal, then the two other numbers are:

If \( \sum_{k=1}^{n} k(k+1)(k-1) = pn^4 + qn^3 + tn^2 + sn \), where \( p, q, t, s \) are constants, then the value of \( s \) is equal to:

Let $S$ be the set of all values of $a_1$ for which the mean deviation about the mean of $100$ consecutive positive integers $a_1, a_2, a_3, \ldots , a_{100}$ is $25$ Then $S$ is

The sum $\displaystyle\sum_{n=1}^{21} \frac{3}{(4 n-1)(4 n+3)}$ is equal to

Let a_n be the n^th term of the series 5 + 8 + 14 + 23 + 35 + 50 +.... and Sn = \(\displaystyle\sum_{k=1}^{n} a_k.\) Then S³⁰ - a⁴⁰ is equal to

If the sum of the series \(\left(\frac{1}{2}-\frac{1}{3}\right) + \left(\frac{1}{2^2}-\frac{1}{2·3} + \frac{1}{3^2}\right) + \left(\frac{1}{2^2}-\frac{1}{2^2·3} + \frac{1}{2·3^2} - \frac{1}{3^3}\right) + \left(\frac{1}{2^4}-\frac{1}{2^3·3} + \frac{1}{2^2·3^2} - \frac{1}{2·3^3} + \frac{1}{3^4}\right) + ........ \)
is \(\frac{α}{β}\) , where α and β are co-prime, then α+3β is equal to ________

\( \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n+1} \) is equal to

The sum of the series \(0.2 + 0.004 + 0.00006 + 0.0000008 + \dots\) is

\[ \sum_{r=1}^{\infty} \left(3\cdot 2^{-r} - 2\cdot 3^{1-r}\right) \]

\[ 1 + \frac{3}{1!} + \frac{5}{2!} + \frac{7}{3!} + \cdots \infty \]

\(\frac{1}{\log_2 10} + \frac{1}{\log_4 10} + \frac{1}{\log_8 10} + \frac{1}{\log_{16} 10}\) is equal to 11

\(\log x + \log x^3 + \log x^5 + \dots + \log x^{2n-1}\) is equal to

If \(a, b\) and \(c\) are in HP, then for any \(n \in \mathbb{N}\), which one of the following is true?