For natural number n, an = 1+1+3+--+--log n . x dt By use log x = hen x > 0,· w 1 t Prove that the series is converg...
(8) Prove that dt= 1-t n=1 for x e [-a, a],0< a< 1 and deduce from there a power series expansion for -In(1-x) (8) Prove that dt= 1-t n=1 for x e [-a, a],0
3) Let F(x) = {* In In(1+t) dt. t (a) Find the Maclaurin series for F: (b) Use the series in part (a) to evaluate F(-1) exactly and use the result to state its interval of convergence. (c) Approximate F(1) to three decimals. (Hint: Look for an alternating series. )
Prove that for each natural number n 26 we have 2n 3 3 2" Use the above to prove that for each natural number n 2 6 we have (n +1)2 Hint: n24n +4-(n2 +2n +1) + (2n+3).] 2" Prove that for each natural number n 26 we have 2n 3 3 2" Use the above to prove that for each natural number n 2 6 we have (n +1)2 Hint: n24n +4-(n2 +2n +1) + (2n+3).] 2"
Let 4. ) Using only the definition of infinite Series convergence, prove the following: w, ZER. Given of in and on respectively convergent to X and Y, then zyn =wX t zY In are 2 WXN t DE 6 Use the theorem above to prove the following: Let WEIR. Given to and are respectively convergent to X and Y, then £ w x n = wX,
Prove this using the definition R7: log(n*) is O(log n) for any fixed x > 0
3. (25 Points) Find f(t). f(0) + f(t - 1)f(t)dt = t. Hint: The second term on the left side is a convolution and it might be helpful to use the Laplace Transform. 1 4. (10 Points) Solve the initial value problem by Laplace transform techniques. x" + 5x' + 4x = 0;x(0) = 1,x'(0) = 0. I 5. (15 Points) Find a series solution for the following differential equation. Calculate the radius of convergence. 2(x - 1)y' = 3y...
Prove the given definition, for parts a) through c). Lemma 9.3.5 (Orthogonality Lemma). Fir N and let w-wN-e2mi/N be the natural primitive Nth root of unity in C. Fort Z/(N), we have: N-1 ktN ift-0 (mod N), 0 otherwise. Lukt (9.3.5) k-0 9.3.2. (Proves Lemma 9.3.5) Fix N є N, and let w-e2m/N. Let f(x)-r"-1. o510 (a) Explain why N-1 (9.3.9) (Suggestion: Try writing out the sum as 1 +z+....) (b) Explain why for any t є z/(N), fw)-0. (c)...
1. Show the series convergent or not. (-1)" (In 2)" n=0 2. Use the root test for the series convergent or not. ~ n2 E (1-5) n=1 3. (x + 1)" 3n Examine the convergence of the power series. Find the convergence radius R and the convergence range. n=1
Let S f(w)dt = 6, f(x)dx = -4, log(x)dt = 12, 9(x) dx = 9 Use these values to evaluate the given definite integral: -3 (f(x) f(x) + g(x)) dx
1. The natural logarithm of (x > 0) can be computed using In(x) dt. Use (a) the mid-point rule, (b) trapezoidal rule, and (c) Simpson's Rule with N 6 subdivisions to approximate In(7) To aid the computation process it might be useful to set up a table containing values for xư x-f(x), f(x), and the weightings for the each of the numerical techniques. For example, i | zi | f(zi) | ที่ | f(r) | midpoint | trapezoidal Simpson's 1...