#31 21. (a) What does it mean for a series to be absolutelyconvergent?(b) What does it mean for a series to be conditionallydi convergent?59(c) If the series of positive terms 2n-1 bn converges, thenwhat can you say about the series -(~4)"6,722-34 Determine whether the series is absolutely synvergent,conditionally convergent, or divergent.23. ΣB22. Σįn=1B24. Σ(-1)"+1n=0B26. Σn=1(-1)"nª28. Σn=1-nin² + 1125sin n2"30. Σ (-1)"-1.n=1832. Σ (−1)"n=134. Σ (-1)"n=2 n ln n35. Σn=1 n!n² + 138. Σn=1B39. ΣΣn² mabnrn² + 4√n³ + 2(-0.8)"(-3)"nsuo vistulnnh(-1)"+¹no(-1)"-1n² 2n25. ΣB40. Σ (-)827. Σn=1(-1)" diyan=1 5n + 1829. Σn=131. Σ33. Σ35-36 Graph both the sequence of terms and the sequence ofpartial sums on the same screen. Use the graph to make a roughestimate of the sum of the series. Then use the AlternatingSeries Estimation Theorem to estimate the sum correct to fourdecimal places.(error] < 0.0005)(-1)"-1√√n²(error] < 0.00005)COS NTn=1 3n + 2100 aproape(-1)"n² + 137-40 Show that the series is convergent. How many terms ofthe series do we need to add in order to find the sum to theindicated accuracy?37. Σ|(error < 0.0005)1 + 2 sin nn³mm36. Σ (−1)"-1.n=1(error] < 0.00005)(-1)" sb ni ufleau voyln nHoton8"SECTION 11.5 Alternating Series and Absolute Convergence41-44 Approximate the sum of the series correct to fourdecimal places.GO41. Σ(-1)"R-1 (2n)!43. Σ (−1)"ne-2,n=145. Is the 50th partial sum 550 of the alternating series1201001 bg-(-1)-/n an overestimate or an underestimate of thetotoong sifT apg8111001113846. Σ109 510total sum? Explain.5646-48 For what values of p is each series convergent?(-1)-1пр47. Σ(-1)"n=1 n + pand therefore42. ΣCO44. ΣΣn=1 1an=849. Show that the series Σ(-1)"¹bn, where bn = 1/n if n is oddand bn = 1/n² if n is even, is divergent. Why does the Alter-nating Series Test not apply?50. Use the following steps to show that(-1)"-1nan + |an|2(-1)^+1по1(-1)"-1n4"48.Σ (-1)"-1n=2= In 2Let h, and sn be the partial sums of the harmonic and alter-nating harmonic series.(a) Show that S2n=h2n- hn.(b) From Exercise 11.3.46 we have1946hn- ln n→Yh2n - ln(2n) → YUse these facts together with part (a) to show thatS2n→ In 2 as n → ∞.as n → ∞(In n)Pas n ∞51. Given any series an, we define a series Zat whose termsare all the positive terms of Σa, and a series Σan whoseterms are all the negative terms of Σan. To be specific, we letan =n-an = |an|2Notice that if an> 0, then a = an, and an = 0, whereas ifan<0, then an an and a = 0.773(a) If Σa, is absolutely convergent, show that both of theseries Eat and Σa, are convergent.(b) If Σa, is conditionally convergent, show that both ofthe series Σa, and Σa are divergent.

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28. IM. IM. M. sin n 2" 30. (-1)-1. 32. (-1)". (-1)" n-2 n ln n 34. Σ n n² + 4 n √n³+2 Tod 29. 31. 33. n=1 1 + 2 sin n (-1)" n-1 n+p COS NTT n-1 3n+2 47. Σ 48. Σ(-1)-1 n-2 (In n) (-1)" sb ni tuleau vipy 49. Show that the series Σ(-1)"bn, where b = 1/n if n is odd and bn = 1/n² if n is even, is divergent. Why does the Alter- nating Series Test not apply? n-2 ln n 50. Use the following steps to show that (-1)"-¹ Σ n=1 n = In 2 n e sole Let h.. and s.. be the partial sums of the harmonic and alter-