Can you help me with questions 9 and 19 from the first picture? Thanks so much!

I AT&T S 12:58 AM 25% SM CHAPTER 10 INFINITE SERIES Solution This series converges absolutely because taking the absolute value of Back term, we obtain a p-series with p = 2 > 1: (convergent p-series) The next theorem tells us that if the series of absolute values converges. then original series also converges. THEOREM 1 Absolute Convergence Implies Convergence If _ lani converges. then a. also converges. Proof We have -ja,| $ a. a, is conditionally convergent. DEFINITION Conditional Convergence An infinite series _ a, converges condi- tionally if _ an converges but lan| diverges. If a series is not absolutely convergent, how can we determine whether it is confi- tionally convergent? This is often a difficult question. because we cannot use the Integral Test or the Direct Comparison Test since they apply only to positive series. However. convergence is guaranteed in the particular case of an alternating series al : An atommating series with [(-1y-b, = by - bit by - by t ... thining terras. The sora is the signed ta, which is at mod by. where the terms b, are positive and decrease to zero (Figure 1). 3 /5\fAT&T S 12:58 AM 24% CHAPTER 10 INFINITE SERIES N-de Therefore, lim Sy = S and the infinite series converges to S. From the incqualm in (1) we also see that 0 999, SN approximates S with error less than 10-3. 5 /5