Let z n 1 z n curren q 1, so that the series z 1 z 2 middot middot middot converges by the ratio test Show that the remainder R n z n 1 z n 2 middot middot middot satisfies the inequality R n curren z n 1 (1 q) Using this, find how many terms suffice for computing the sum s of the series i 2 n 1

Question: Let |z n+1 /z n | ¤ q < 1, so that the series z 1 + z 2 + · · · converges by

Let |z_n+1/z_n| ‰¤ q < 1, so that the series z₁+ z₂+ · · · converges by the ratio test. Show that the remainder R_n= z_n+1+ z_n+2+ · · · satisfies the inequality |R_n| ‰¤ |z_n+1|/(1 - q). Using this, find how many terms suffice for computing the sum s of the series

:+ i п 2

with an error not exceeding 0.05 and compute s to this accuracy.

:+ i 2"n -1

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