Find smallest n so that the finite sum ∑ k = 1 n ( − 1 ) k k 4 k is certain to differ from ∑ k = 1 ∞ ( − 1 ) k k 4 k by less than 0.002.

Question

Find smallest n so that the finite sum       ∑          k      =      1              n                  (      −      1              )                  k                    k              4              k             is certain to differ from       ∑          k      =      1              ∞                  (      −      1              )                  k                    k              4              k             by less than 0.002.

kunstboom8w · Accepted Answer

Since the series is decreasing, you just have to find k forthe term which equals 0.002.  0.002  =  k      /        4          k            4          k        =  500  kyou cant solve this directly, but using numeric methods, you findk =   5.744So the smallest n valueis 5, because the sum from 6 to   ∞ is guaranteed to be less than 0.002.This is because the term for k = 6 is less than 0.002 and the alternating decreasing sequence guarantees that the sum of the remainder of the sequence will be less than this also.

Find smallest n so that the finite sum sum_(k=1)^{n}(-1)^(k)k/(4^k) is certain to differ from sum_(k=1)^(infty)((-1)^(k)k)/(4^k) by less than 0.002.

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