(Summation by parts) Let fung and fvng be two sequences, and

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(Summation by parts) Let fung and fvng be two sequences, and let sn D Pn kD1 vk. (a) Show that Pn kD1 ukvk D unC1snCPn k

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(Summation by parts) Let fung and fvng be two sequences, and let sn D Pn kD1 vk. (a) Show that Pn kD1 ukvk D unC1snCPn kD1.uk?ukC1/sn. (Hint: Write vn D sn ? sn?1, with s0 D 0, and rearrange the sum.) (b) If fung is positive, decreasing, and convergent to 0, and if fvng has bounded partial sums, jsnj K for all n, where K is a constant, show that P1 nD1 unvn converges. (Hint: Show that the series P1 nD1.un?unC1/sn converges by comparing it to the telescoping series P1 nD1.un ? unC1/.)

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