Question: Determine whether the series is convergent or

> 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 +

> Find the radius of convergence and interval of convergence of the series. (4x + 1)" n? 00 n-1

> Determine whether the series is convergent or divergent. If it is convergent, find its sum. Σ 2 arctan n

> Determine whether the series is convergent or divergent. If it is convergent, find its sum. -1

> Determine whether the series is convergent or divergent. If it is convergent, find its sum. 1+ 3" 24

> Determine whether the series is convergent or divergent. If it is convergent, find its sum. 1+ 24 Σ 34

> Determine whether the series is convergent or divergent. If it is convergent, find its sum. 00 Σ cos A-1

> Determine whether the series is convergent or divergent. If it is convergent, find its sum. k? Σ k? – 1 00 - 1 k-2

> Determine whether the series is convergent or divergent. If it is convergent, find its sum. k(k + 2) Σ -i (k + 3)2 k-1

> Explain what it means to say that ∑∞n-1 an = 5.

> Determine whether the series is convergent or divergent. If it is convergent, find its sum. n - 1 Σ A-1 3n Зп — 1

> Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? E (-1)*-'ne- (1 error|< 0.01)

> Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. Σ n-0

> Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 00 Σ A-o 3*+1

> Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 10" Σ (-9)*-1

> Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 00 E 6(0.9)“-! A-1 n-1

> Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 1 + 0.4 + 0.16 + 0.064 +

> Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 10 – 2 + 0.4 – 0.08 + ·

> Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 4 + 3 ++ + ...

> Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 3 – 4 +4 - + ... 16

> (a). Explain the difference between (b). Explain the difference between aj and 内I and E aj

> (a). What is the difference between a sequence and a series? (b). What is a convergent series? What is a divergent series?

> Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? (-1)*+1 (lerror|< 0.00005) A-1

> (a). What is an alternating series? (b). Under what conditions does an alternating series converge? (c). If these conditions are satisfied, what can you say about the remainder after n terms?

> On what interval is the curve y = fx0 t2/t2 + t + 2, dt concave downward?

> If f (x) = fsinxo√1 + t2 dt and g (y) = fy3 f(x) dx, find g"(π/6).

> If f (x) = fx0 (1 – t2) et2 dt, on what interval is f increasing?

> Evaluate the integral. f01 10x dx

> Evaluate the integral. f12 x-2 dx

> Evaluate the integral. f1/2√3/2 6/√1 – t2 dt

> find f. f'(x) = √x (6 + 5x) f (1) = 10

> find f. f'(x) = 8x3 + 12x + 3, f (1) = 6

> Find f. f"(x) = 1 - 6x, f (0) = 8

> Find f. f"(x) = 6x + sin x

> Find f. f"(x) = 6x + 12x2

> Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. *cos X y = * (1 + v?)1º dv Jsin x

> Find the antiderivative F of f that satisfies the given condition. Check your answer by comparing the graphs of f and F. f(x) — 4 — 3(1 + х?)-1, F(1) = 0

> Find the antiderivative F of f that satisfies the given condition. Check your answer by comparing the graphs of f and F. f(x) = 5x* – 2x', F(0)=4

> Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = 27 + 6 cos x COS %3D

> Find the most general antiderivative of the function. (Check your answer by differentiation.) g(8) = cos e – 5 sin e

> Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = 3e* + 7 sec?x

> Let S be the solid obtained by rotating the region shown in the figure about the y-axis. Sketch a typical cylindrical shell and find its circumference and height. Use shells to find the volume of S. Do you think this method is preferable to slicing? Expl

> Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on [0, 1]. 2 3 lim +

> Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on [0, 1]. lim 4

> The area labeled B is three times the area labeled A. Express a in terms of b. y. y= e" y= e" B A a

> Suppose h is a function such that h (1) = -2, h'(1) = 2, h"(1) = 3, h (2) = 6, h'(2) = 5, h"(2) = 13, and h" is continuous everywhere. Evaluate f21 h"(u) du.

> A manufacturer of corrugated metal roofing wants to produce panels that are 28 in. wide and 2 in. thick by processing flat sheets of metal as shown in the figure. The profile of the roofing takes the shape of a sine wave. Verify that the sine curve has e

> (a). If f is continuous, prove that fπ/20 f (cos x) dx = fπ/20 f (sin x) dx (b). Use part (a) to evaluate fπ/20 cos2 x dx and x fπ/20 sin 2x dx.

> If a and b are positive numbers, show that r(1 – x)* dx = [ x*(1 – x)* dx

> If f is continuous on R, prove that fbaf (x + c) dx = fb+ca+c f (x) dx For the case where f (x) > 0, draw a diagram to interpret this equation geometrically as an equality of areas.

> If f is continuous on R, prove that fbaf (-x) dx = f-1-bf (x) dx For the case where f (x) > 0 and 0 < a < b, draw a diagram to interpret this equation geometrically as an equality of areas.

> If f is continuous and f90 f (x) dx = 4, find f30 xf (x2) dx.

> If f is continuous and f40f (x) dx = 10, find f20f (2x) dx.