Let a n be a series with positive terms and let r n = a n+1 /a

Let Σa_n be a series with positive terms and let r_n = a_n+1/a_n . Suppose that lim_n→∞ r_n = L < 1, so Σa_n converges by the Ratio Test. As usual, we let Rn be the remainder after n terms, that is,

R_n = a_n+1 + a_n+2 + a_n+3 + ∙ ∙ ∙

(a) If {r_n} is a decreasing sequence and r_n+1< 1, show, by summing a geometric series, that

(b) If {r_n} is an increasing sequence, show that

Transcribed Image Text:

an+1 Ra 1 - - ra+1