Column-I Column I. If the first term of an infinite G.P. is...

Column-I Column I. If the first term of an infinite G.P. is 1 and each term is twice the sum (A) $\frac{2}{9}$ of the suceeding terms, then the common ratio is II. Sum to infinity of the series $\frac{2}{3}-\frac{5}{6}+\frac{2}{3}-\frac{11}{24}+\ldots$ is (B) $\frac{3}{2}$ III. $\lim _{n \rightarrow \infty}\left(1+3^{-1}\right)\left(1+3^{-2}\right)\left(1+3^{-4}\right)\left(1+3^{-8}\right) \ldots\left(1+3^{-2^{\prime}}\right)=$ (C) 1 IV. If $\sum_{k=1}^{n}\left(\sum_{m=1}^{k} m^{2}\right)=a n^{4}+b n^{3}+c n^{2}+d n+e$, then $a+b+c+d+e=$ (D) $\frac{1}{3}$

Column-I Column
I. If the first term of an infinite G.P. is 1 and each term is twice the sum
(A) $\frac{2}{9}$
of the suceeding terms, then the common ratio is
II. Sum to infinity of the series $\frac{2}{3}-\frac{5}{6}+\frac{2}{3}-\frac{11}{24}+\ldots$ is
(B) $\frac{3}{2}$
III. $\lim _{n \rightarrow \infty}\left(1+3^{-1}\right)\left(1+3^{-2}\right)\left(1+3^{-4}\right)\left(1+3^{-8}\right) \ldots\left(1+3^{-2^{\prime}}\right)=$
(C) 1
IV. If $\sum_{k=1}^{n}\left(\sum_{m=1}^{k} m^{2}\right)=a n^{4}+b n^{3}+c n^{2}+d n+e$, then $a+b+c+d+e=$
(D) $\frac{1}{3}$

Key Concepts

Double Summation and Extraction of Polynomial Coefficients

Double summation involves summing over two indices and often results in expressions that can be represented as polynomials in n. A key insight is that the sum of the coefficients of a polynomial is equivalent to evaluating the polynomial at n = 1. This method is particularly useful when dealing with nested sums and requires familiarity with summation formulas, such as the sum of squares or higher powers, and techniques for simplifying polynomial expressions.

Infinite Product and Convergence

Infinite products involve the multiplication of an infinite number of factors and their limit as the number of terms tends to infinity. Analyzing such products requires understanding convergence criteria and the manipulation of exponents and series to compute the limit. This concept is important when evaluating the behavior of products that can be written in forms that lead to telescoping or summing techniques.

Sum to Infinity of a Geometric Series

The sum to infinity of a geometric series is the sum of all the terms in a convergent infinite geometric progression, which is calculated using the formula S = a/(1-r), where a is the first term and r is the common ratio. This concept is crucial in determining the limit of the series as the number of terms approaches infinity and applies under the condition that |r| < 1.

Infinite Geometric Progression

This concept deals with sequences where each term after the first is found by multiplying the previous term by a constant called the common ratio. An infinite geometric progression converges if the absolute value of the common ratio is less than one. Understanding the properties of such sequences, including how changes in the first term or common ratio affect the progression, is fundamental to solving various problems in series and sequences.

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