Question: Estimate ∑∞n=1 (2n + 1)-6 correct

> Find the radius of convergence and interval of convergence of the series. 00 n!

> Suppose that ∑∞n=0 cnxn converges when x = -4 and diverges when x= 6. What can be said about the convergence or divergence of the following series? ( a ) Σ ca A-0 (b) E ca n-0 ( c) Σ C-3)" ( d) Σ (-1)'c, 9" A-0 A-

> Find the radius of convergence and interval of convergence of the series. 00 E yn x"

> Test the series for convergence or divergence. ∑∞n=1 (-1)n-1 / 2n + 1

> Test the series for convergence or divergence. -3/4 + 5/5 – 7/6 + 9/7 – 11/8 + …

> Test the series for convergence or divergence. 4/7 – 4/8 + 4/9 – 4/10 + 4/11 - ….

> Suppose f is a continuous positive decreasing function for x > 1 and an = f (n). By drawing a picture, rank the following three quantities in increasing order: 5 6 °f(x) dx Σ 2 ai i-l i-2

> Draw a picture to show that What can you conclude about the series? Σ 1.3 -2 n 1.3

> Test the series for convergence or divergence. ∑∞n=1 (-1)n+1 n/n2 + 9

> Test the series for convergence or divergence. ∑∞n=1 (-1)n-1 / 2n + 1

> Test the series for convergence or divergence. ∑∞n=1 (-1)n-1 / ln (n + 4)

> The terms of a series are defined recursively by the equations Determine whether ∑an converges or diverges. 5n + 1 aj = 2 An+1 an 4n + 3

> If ∑∞n=0 cn4n is convergent, does it follow that the following series are convergent? (a) 2 ca(-2)ª (b) Σ c.(-4)" A-0 A-0

> What can you say about the series ∑an in each of the following cases? An+1 an+1 (а) lim an = 8 (b) lim = 0.8 an+1 (c) lim

> Determine whether the series is absolutely convergent. 2/5 + 2 ∙ 6/5 ∙ 8 + 2 ∙ 6 ∙ 10/5 ∙ 8 ∙ 11 + 2 ∙ 6 ∙ 10 ∙ 14/5 ∙ 8 ∙ 11 ∙ 14 + ∙∙∙

> Determine whether the series is absolutely convergent. 1 – 1 ∙3/3! + 1∙3∙5/5! - 1∙2∙5∙7/7!+ …+ (-1)n-1 1∙3∙5∙∙∙∙(2n – 1)/(2n – 1)!+ ∙∙∙

> Determine whether the series is absolutely convergent. ∑∞n=1 (-2)n n!/(2n)!

> Determine whether the series is absolutely convergent. ∑∞n=1 (-1)n arctan n/n2

> Determine whether the series is absolutely convergent. ∑∞n=1 (-1)n-1/√n

> Determine whether the series is absolutely convergent. ∑∞n=1 (-1)n-1 √n/n + 1

> Determine whether the series is absolutely convergent. ∑∞n=0 (-10)n/n!

> Determine whether the series is absolutely convergent. ∑∞n=1 n!/1003

> Determine whether the series is absolutely convergent. ∑∞n=1 (-3)n/n3

> Approximate the sum of the series correct to four decimal places. 00 (-1)* オー」 3"n!

> Find the radius of convergence and interval of convergence of the series. n?x" Σ 2.4. 6. ...• (2n)

> Approximate the sum of the series correct to four decimal places. (-1)*-'n? 00 10"

> Graph both the sequence of terms and the sequence of partial sums on the same screen. Use the graph to make a rough estimate of the sum of the series. Then use the Alternating Series Estimation Theorem to estimate the sum correct to four decimal places.

> Graph both the sequence of terms and the sequence of partial sums on the same screen. Use the graph to make a rough estimate of the sum of the series. Then use the Alternating Series Estimation Theorem to estimate the sum correct to four decimal places.

> Determine whether the series is convergent or divergent. ∑∞n=1 n2/n3 + 1

> Determine whether the series is convergent or divergent. ∑∞n=1 ne-n

> Determine whether the series is convergent or divergent. 1 + 1/2√2 + 1/3√3 + 14√4 + 1/5√5+ …

> For what values of p is the following series convergent? (-1)ª-1 00 nº

> Calculate the first 10 partial sums of the series and graph both the sequence of terms and the sequence of partial sums on the same screen. Estimate the error in using the 10th partial sum to approximate the total sum. (-1)^-1 ,3 ー1

> Is the 50th partial sum s50 of the alternating series ∑∞n=1(-1)n-1/n an overestimate or an underestimate of the total sum? Explain.

> Test the series for convergence or divergence. ∑∞n=1 (-1)n cos (π/n)

> Find the radius of convergence and interval of convergence of the series. x" Σ 1:3. 5. (2n – 1)

> Use the Comparison Test to determine whether the series is convergent or divergent. n Σ 2n' + 1 ,3 A-1

> Use the Integral Test to determine whether the series is convergent or divergent. 1 Σ In + 4 00 A-l V

> Use the Integral Test to determine whether the series is convergent or divergent. In

> Use the Integral Test to determine whether the series is convergent or divergent. 00 Σ ,5 A-1 n

> It is important to distinguish between and What name is given to the first series? To the second? For what values of does the first series converge? For what values of does the second series converge? Σ E nº and A-1 A-1

> Show that if an > 0 and ∑an is convergent, then ∑ ln (1 + an) is convergent.

> The meaning of the decimal representation of a number 0.d1d2d3.. (where the digit di is one of the numbers 0, 1, 2, . . . , 9) is that O.d,dzdzd4... di dz d3 da 10 104 10 102

> Suppose ∑an and ∑bn are series with positive terms and ∑bn is known to be divergent. (a). If an > bn for all n, what can you say about ∑an? Why? (b). If an < bn for all n, what can you say about ∑an? Why?

> (a). Use a graph of y = 1 to show that if sn is the nth partial sum of the harmonic series, then (b). The harmonic series diverges, but very slowly. Use part (a) to show that the sum of the first million terms is less than 15 and the sum of the first b

> Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. sin'n 00 n-1

> Find the radius of convergence and interval of convergence of the series. 00 2n Σ a-2 n(In n)?

> Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. 1 Σ Vn' + 1 00

> How many terms of the series ∑∞n=1 1/ [n (ln n)2] would you need to add to find its sum to within 0.01?

> Find the sum of the series ∑∞n=1 1/n5 correct to three decimal places.

> (a). Use the sum of the first 10 terms to estimate the sum of the series∑∞n=1 1/n2. How good is this estimate? (b). Improve this estimate using (4) with n = 10. (c). Find a value of that will ensure that the error in the approximation s ≈ sn is less than

> (a). Find the partial sum s10 of the series∑∞n-1 1/n4. Estimate the error in using s10 as an approximation to the sum of the series. (b). Use (4) with n = 10 to give an improved estimate of the sum. (c). Find a value of n so that sn is within 0.00001 of

> Find the values of p for which the following series is convergent. 1 Σ -2 n(ln n)P 00

> Determine whether the series is convergent or divergent. ∑∞n=1 n2 – 5n/n3 + n + 1

> List the first six terms of the sequence defined by an = n/2n + 1 Does the sequence appear to have a limit? If so, find it.

> Determine whether the series is convergent or divergent. ∑∞n=1 sin (1/n)

> Find the radius of convergence and interval of convergence of the series. n (x – a)", b>0

> Determine whether the series is convergent or divergent. ∑∞n=0 1 + sin n/10n

> Determine whether the series is convergent or divergent. ∑∞n=1 2 + (-1)n n √n

> Determine whether the series is convergent or divergent. If it is convergent, find its sum. Σ (cos 1 ) k-1

> Determine whether the series is convergent or divergent. ∑∞n=1 1 + 4n/1 + 3n

> Determine whether the series is convergent or divergent. 1/5 + 1/8 + 1/11 + 1/14 + 1/17+ …

> Determine whether the series is convergent or divergent. 1 + 1/3 + 1/5 + 1/7 + 1/9 + …

> Determine whether the series is convergent or divergent. ∑∞n=1 4 + 3n/2n

> Determine whether the series is convergent or divergent. ∑∞n=1 n – 1/n4n

> Determine whether the series is convergent or divergent. ∑∞n=1 n2 – 1/3n4 + 1

> Determine whether the series is convergent or divergent. ∑∞n=1 cos2n/n2 + 1

> Determine whether the sequence converges or diverges. If it converges, find the limit. (-1)*n n + 2n² + 1

> Determine whether the series is convergent or divergent. ∑∞n=1 1/n2 + 9

> Determine whether the series is convergent or divergent. ∑∞n=2 1/n ln n

> Determine whether the series is convergent or divergent. 1 + 1/8 + 1/27 + 1/64 + 1/125 + . . .

> Determine whether the series is convergent or divergent. ∑∞n=1 (n-14 + 3n-12)

> Determine whether the series is convergent or divergent. ∑∞n=1 2/n0.85

> Use the Comparison Test to determine whether the series is convergent or divergent. 00 .3 Σ n² – 1

> Let an = 2n/3n + 1 (a). Determine whether {an} is convergent. (b). Determine whether ∑∞n-1 an is convergent.

> Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain w

> Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain w

> Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain w

> Find the radius of convergence and interval of convergence of the series. E n!(2x 1)"

> Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain w

> Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. {5, 1, 5, 1, 5, 1, . ..}

> (a). Determine whether the sequence defined as follows is convergent or divergent: (b). What happens if the first term is a1 = 2? a, = 1 an+1 = 4 - a, for n>1

> We have seen that the harmonic series is a divergent series whose terms approach 0. Show that is another series with this property. 00 2 In( 1 + A-1

> Find the values of x for which the series converges. Find the sum of the series for those values of x. 00 cos"x Σ 24 n-0

> Find the values of x for which the series converges. Find the sum of the series for those values of x. (х + 3)" Σ 24 A-0

> Find the values of x for which the series converges. Find the sum of the series for those values of x. 3" n-

> Express the number as a ratio of integers. 7. ¯12345

> Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain w

> Express the number as a ratio of integers. 1. ¯5342

> Find the radius of convergence and interval of convergence of the series. - 4)ª n(x – 4)" n + 1 A-1

> Express the number as a ratio of integers. 6. ¯254 = 6.2545454…

> Express the number as a ratio of integers. 0. ¯2 = 0.2222…

> Express the number as a ratio of integers. 0. ¯73 = 0.73737373…

> Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. a, = 1 + (-2/e)"

> Determine whether the series is convergent or divergent by expressing Sn as a telescoping sum (as in Ex am ple 6). If it is convergent, find its sum. 00 E In- п+1 A-1

> Determine whether the series is convergent or divergent by expressing Sn as a telescoping sum (as in Ex am ple 6). If it is convergent, find its sum. 3 Σ п(n + 3) A-1

> Determine whether the series is convergent or divergent by expressing Sn as a telescoping sum (as in Ex am ple 6). If it is convergent, find its sum. 00 2 Σ A-1 n + 4n + 3

> Determine whether the series is convergent or divergent by expressing Sn as a telescoping sum (as in Ex am ple 6). If it is convergent, find its sum. 2 Σ n² – 1 ,2

> Determine whether the sequence converges or diverges. If it converges, find the limit. sin 2n 1 + + Jn

> Determine whether the series is convergent or divergent. If it is convergent, find its sum. 1 Σ п(n + 1) A-1 +