Sum to n terms of the series (1)/(5 !)+(1 !)/(6 !)+(2 !)/(7 !)+(3 !)/(8 !)+…is (A) (2)/(5 !)-(1)

step 1 So, in this question, we have TR is equal to R minus 1 factorial, divided by R plus 2, divided b

Sum to n terms of the series (1)/(5 !)+(1 !)/(6 !)+(2 !)/(7...

Sum to $n$ terms of the series $\frac{1}{5 !}+\frac{1 !}{6 !}+\frac{2 !}{7 !}+\frac{3 !}{8 !}+\ldots$ is (A) $\frac{2}{5 !}-\frac{1}{(n+1) !}$ (B) $\frac{1}{4}\left(\frac{1}{4 !}-\frac{n !}{(n+4) !}\right)$ (C) $\frac{1}{4}\left(\frac{1}{3 !}-\frac{3 !}{(n+2) !}\right)$ (D) None of these

Sum to $n$ terms of the series $\frac{1}{5 !}+\frac{1 !}{6 !}+\frac{2 !}{7 !}+\frac{3 !}{8 !}+\ldots$ is
(A) $\frac{2}{5 !}-\frac{1}{(n+1) !}$
(B) $\frac{1}{4}\left(\frac{1}{4 !}-\frac{n !}{(n+4) !}\right)$
(C) $\frac{1}{4}\left(\frac{1}{3 !}-\frac{3 !}{(n+2) !}\right)$
(D) None of these

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Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is the process of breaking down a complex rational function into a sum of simpler fractions. This method is useful when dealing with series terms that are expressed as ratios of factorials or other functions, as it can expose a telescoping behavior that simplifies the evaluation of the series' sum.

Index Shifting

Index shifting is a technique used to re-index a series to make its pattern or cancellation properties more apparent. By redefining the index of summation, one can often reveal a telescoping structure or simplify complex expressions involving factorials.

Telescoping Series

A telescoping series is one in which consecutive terms cancel one another out when summed, leaving only a few terms from the beginning and end of the series. This property can be exploited to simplify the summation process and derive a closed-form expression for the sum of the series.

Factorial Properties

This concept involves understanding the definition and properties of the factorial function. In many series, factorial expressions can often be simplified by canceling common terms in the numerator and denominator. Recognizing that n! can be written as n · (n-1)! is key to rewriting and simplifying the terms in a series that contain factorials.

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