Use the formula for the sum of a geometric series to find the sum. ∑n=4∞(−49)n

Question

Use the formula for the sum of a geometric series to find the sum.  ∑n=4∞(−49)n

Pohanginah · Accepted Answer

Given series:  ∑n=4∞(−49)n  For the given series the ratio of (n+1)-th term to n-th term is constant which is −49  Hence, the terms of the given series are in geometric progression.  Write the given series in standard geometric series form.  That is,  ∑n=4∞(−49)n=(−49)−4+(−49)−3+(−49)−2+(−49)−1+∑n=0∞(−49)n  ∑n=4∞(−49)n=4365256+∑n=0∞(−49)n  The general geometric series,  ∑n=0∞arn, where a is the first term and r is the common ratio   converges to a1−r, for -1&amp;lt;r&amp;lt;1  Otherwise it diverges.  Compare the series ∑n=0∞(−49)n with the general geometric series, to get  a=1,r=−49∫(−1,1)  Hence, ∑n=0∞(−49)n=11−(−49)  =11+49  =1(139)  =913  Plug ∑n=0∞(−49)n=913 in equation (1), to get  ∑n=−4∞(−49)n=4365256+913  =590493328  ≈17.7431  Thus,  ∑n=−4∞(−49)n=17.7431

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

Use the formula for the sum of a geometric series to find the sum. sum_{n=4}^infty(-frac49)^n

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