Question:
(a) Use a computer algebra system to evaluate the following integrals.
(b) Based on the pattern of your responses in part (a), guess the value of the integral
if a ≠b. What if a = b?
(c) Check your guess by asking your CAS to evaluate the integral in part (b). Then prove it using partial fractions.
Transcribed Image Text:
1 1 (i) f dx (x + 2)(x + 3) (ii) (x + 1)(x + 5) dx 1 (iii) | dx (x + 2)(x – 5) dx (iv) (x + 2)* 1 dx (x + a)(x + b)
> Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen. f(a) — хе 2", а — 0 ()
> 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
> 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
> 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
> Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent? (-1)* "-| n!
> Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent? Σ sinn R-1
> Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent?
> Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent? 1 Σ i n* + n?
> Use a calculator to find the length of the curve correct to four decimal places. If necessary, graph the curve to determine the parameter interval. r = tan 0, 7/6 < 0 < T/3
> he outer circle in the figure has radius 1 and the centers of the interior circular arcs lie on the outer circle. Find the area of the shaded region.
> (a) What is the difference between a sequence and a series? (b) What is a convergent series? What is a divergent series?
> Find all values of c for which the following series converges. n + 1
> Determine whether the series is absolutely convergent or conditionally convergent. sin n 2" n-1
> (a) Use (4) 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 billion terms is l
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. L.2' dr
> (a) Use a computer algebra system to evaluate the following integrals. (b) Based on the pattern of your responses in part (a), guess the value of / Then use your CAS to check your guess. (c) Based on the patterns in parts (a) and (b), make a conjecture a
> (a) Use the sum of the first 10 terms to estimate the sum of the series / How good is this estimate? (b) Improve this estimate using (3) with n = 10. (c) Compare your estimate in part (b) with the exact value given in Exercise 34. (d) Find a value of n t
> Euler also found the sum of the p-series with p = 4: Use Euler’s result to find the sum of the series. IT 3(4) = E n° 90 (a) 2 1 (b) E (k – 2)4 8
> Test the series for convergence or divergence. n? – 1 E (-1)". n2 + 1
> Leonhard Euler was able to calculate the exact sum of the p-series with p = 2: Use this fact to find the sum of each series. 2 IT 3(2) = Σ 6. (a) Σ 1 (b) Σ R-2 n (n + 1)? n-3 1 (c) Σ (2n)?
> The Riemann zeta-function / is defined by and is used in number theory to study the distribution of prime numbers. What is the domain of /?
> Find the values of p for which the series is convergent. In n Σ n-1
> Find the values of p for which the series is convergent. E n(1 + n²)"
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. Sp dp 12
> Find the values of p for which the series is convergent. 1 Σ n In n [In(In n)]P
> Find the values of p for which the series is convergent. 1 Σ n(In n)" -2
> (a) Use a computer algebra system to evaluate the following integrals. (b) Based on the pattern of your responses in part (a), guess the value of (c) Use integration by parts to prove the conjecture that you made in part (b). For what values of n is it v
> Explain why the Integral Test can’t be used to determine whether the series is convergent. CS TN R-1
> Determine whether the series is convergent or divergent. Σ n-1 n* + 1
> Determine whether the series is convergent or divergent. 1 Σ n2 + 3 n-
> Determine whether the series is convergent or divergent. E ke * k-1
> Test the series for convergence or divergence. 2(-1)- n² – 1 n + 1 E(-1) - R-1
> Determine whether the series is convergent or divergent. ke k-1
> Determine whether the series is convergent or divergent. In n -2 n
> Use a Maclaurin series to obtain the Maclaurin series for the given function. f(x) = x² In(1 + x')
> Determine whether the series is convergent or divergent. 1 n-2 n In n
> Determine whether the series is convergent or divergent. Зп — 4 Σ R-3 n 2n
> Determine whether the series is convergent or divergent. 3. Σ -1 n* + 4
> (a) Use a computer algebra system to evaluate the following integrals. (b) Based on the pattern of your responses in part (a), guess the value of the integral (c) Check your guess with a CAS. Then prove it using the techniques of Section 7.2. For what va
> Determine whether the series is convergent or divergent. 1 -1 n? + 2n + 2
> Determine whether the series is convergent or divergent. 1 n2 + 4 8
> Determine whether the series is convergent or divergent. 1 + n³/2
> Determine whether the series is convergent or divergent. Vn + 4 n° ,2
> Determine whether the series is convergent or divergent. 1 + 2/2 1 + 4/4 1 + 3/3 5/5
> Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen. S(x) — sin x, a — п/6
> Determine whether the series is convergent or divergent. 1 1 3 11 15 19
> Determine whether the series is convergent or divergent. 11 13
> Determine whether the series is convergent or divergent. Σ 0.9999 n-3
> Use the Integral Test to determine whether the series is convergent or divergent. E n’e * R-1
> Use the Integral Test to determine whether the series is convergent or divergent. Σ A n² + 1
> Use the Integral Test to determine whether the series is convergent or divergent. 1 Σ (Зл — 1)*
> Use the Integral Test to determine whether the series is convergent or divergent. 2 Σ 5n – 1 ー1
> Use the Integral Test to determine whether the series is convergent or divergent. En 0.3 R-1 8
> Use the Integral Test to determine whether the series is convergent or divergent. En R-1
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 1 · dx (2х + 1)°
> 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 E ai E ai i-2 i-I
> Draw a picture to show that What can you conclude about the series? 1 Σ 1.3 -2 n 1.3
> For values of k between 3.6 and 4, compute and plot at least 100 terms and comment on the behavior of the sequence. What happens if you change p0 by 0.001? This type of behavior is called chaotic and is exhibited by insect populations under certain condi
> Evaluate the integral. tan 2y dy
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 1 3 — 4х 00
> (a) Show that the function defined by is not equal to its Maclaurin series. (b) Graph the function in part (a) and comment on its behavior near the origin. 1/22 if x + 0 f(x) = if x- 0
> 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 Show that this series always converges. di 0.d,d;dzd4 . .. 10 dz dz d4 102 103 104
> Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. 1 -1 3" + 4"
> Experiment with values of k between 3.4 and 3.5. What happens to the terms?
> Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. E 5 "cos’n cos'n 2.
> Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. 1/m nº
> Match the differential equation with its direction field (labeled I–IV). Give reasons for your answer. y' = x(2 – y) II yA --
> Determine whether the sequence converges or diverges. If it converges, find the limit. tan 'n an n
> Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. 1 5 + n*
> Determine whether the series converges or diverges. Σ 1+1/m
> Determine whether the series converges or diverges. Σ sin
> Determine whether the series converges or diverges. n! n"
> Determine whether the series converges or diverges. n! -1
> Explain why the Integral Test can’t be used to determine whether the series is convergent. cos'n Σ + n?
> Determine whether the series converges or diverges.
> Calculate terms of the sequence for a value of k between 3 and 3.4 and plot them. What do you notice about the behavior of the terms?
> Determine whether the series converges or diverges. 1 Σ -2 n Vn? – 1
> Evaluate the integral. (In x)² 2
> Determine whether the series converges or diverges. е" + 1 -1 пе" + 1
> Determine whether the series converges or diverges. n + 3" -i n + 2"
> Determine whether the series converges or diverges. 5 + 2n Σ (1 + n²)² 2)2
> Determine whether the series converges or diverges. n + 2 Σ (n + 1)3 A-3
> Determine whether the series converges or diverges. V1 + n Σ 2 + n n-1
> Determine whether the series converges or diverges. n2 + n + 1 n* + n? 2
> Determine whether the series converges or diverges. n п +1 -i n + n
> Determine whether the series converges or diverges. Σ Vn + 2
> If f(x) = (1 + x3)30, what is f(58)(0)?
> Use the Ratio Test to determine whether the series is convergent or divergent. (-2)" Σ n2 -1
> Evaluate the integral. x² + 2x) cos x dx
> Determine whether the series converges or diverges. 1 Σ Vn² + 1 n-1
> Determine whether the series converges or diverges. i n"
> Determine whether the series converges or diverges. 4"11 Σ 3" – 2
> Determine whether the series converges or diverges. Σ V3n4 + 1
> Determine whether the series converges or diverges. 1 + cos n e"
> Determine whether the series converges or diverges. (2k – 1)(k² – 1) Σ (k + 1)(k² + 4)²
> Determine whether the series converges or diverges. Σ k3 + 4k + 3 k-1
> Determine whether the series converges or diverges. k sin?k 1 1 + k³ WI
> Determine whether the series converges or diverges. In k k k-1
> Determine whether the series converges or diverges. 6" -i 5" - 1
> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. et dx 4 - e2*
