Sequences and Series

The sum of series \({1 \over {2\,!}} + {1 \over {4\,!}} + {1 \over {6\,!}} + ........\) is

The sum of the first n terms of the series \({1^2} + {2.2^2} + {3^2} + {2.4^2} + {5^2} + {2.6^2} + ....\,is\,{{n{{(n + 1)}^2}} \over 2}\) when n is even. When n is odd the sum is

The sum of the serier \({1 \over {1.2}} - {1 \over {2.3}} + {1 \over {3.4}}..............\) up to \(\infty\) is equal to

l, m, n are the \({p^{th}}\), \({q^{th}}\) and \({r^{th}}\) term of a G.P all positive, \(then\,\left| {\matrix{ {\log \,l} & p & 1 \cr {\log \,m} & q & 1 \cr {\log \,n} & r & 1 \cr } } \right|\,equals\)

If 1, \({\log _9}\,\,({3^{1 - x}} + 2),\,\,{\log _3}\,\,({4.3^x} - 1)\) are in A.P. then x equals

The value of \(\,{2^{1/4}}.\,\,{4^{1/8}}.\,{8^{1/16}}...\infty\) is

Fifth term of a GP is 2, then the product of its 9 terms is

Sum of infinite number of terms of GP is 20 and sum of their square is 100. The common ratio of GP is

\({1^3} - \,\,{2^3} + {3^3} - {4^3} + ... + {9^3} =\)