ANSWER
Given
**do not use integral or cauchy condensation 3) Show that for a > 0, C l...
Problem 6) (The Cauchy condensation test] Let {an} be a nonincreasing sequence of positive numbers (an > an+1 for all n) that converges to 0. The Cauchy condensation test states that Dan converges if and only if 2"2n converges. For example, 1/n diverges because 2" (1/2") = 1 diverges. Explain why the test works.
2 11.73 The classical defenition of the exponential integral Eiw for x> 0 B the Cauchy Principal value integral. Ei= ce de where Show result the integration range is cut at xoor that this defenition gields a convergent for positive X.
5(a)(b) are asking what the
Cauchy-Goursat Theorem and the general Cauchy Integral Theorem
talks about. Please use these two theorems to solve the
problem.
(6) Let C denote the closed contour (3 – sint)et, 0 <t < 2n. Use 5(a)(b) above to aid in computing the following contour integrals. (a) So z?sin(2)dz (b) Jc E-P-5)² dz 24-iz
Page < 2 > of 3 ZOOM 3. Use the Integral Test or a Comparison Test to determine if the series shown below converges or diverges. Be sure to check that the conditions of the Integral Test or Comparison Test are satisfied. 4" (sin 4"(sin(n) + 1) 22-1
l. Let wn > 0 and 〈 > 0. Show that s2 + 2(Wns+uậ = 0 has (a) complex roots when 0 < £1. (b) real and equal roots when ς-1, and real and distinct roots when ς > 1
(5) Use Cauchy Integral Formula to calculateh(2+(i+1), ee is whose vertices are 0, 4, 2- 2i, and -2i. oriented counterclockwise. Assume a suitable branch of (z +4i) c (2+ I)2 + i dz where C is the paralle 5)
please show all steps
Find L{f}(s) directly by evaluating the integral if 2t when 0 <t<3, when t > 3.
Let (an) be a sequence such that lim an = 0. Define the sequence (AR) Exercise 21: by A =ļa, and An = zou-a + ax=a + zam for k21. Prove that an converges to some S if and only if Ax converges to S. N=0 k=0 Exercise 22: (Cauchy condensation test) Let (an) be a sequence such that 0 < antı san a) Show n=0 n=1 Hint: Recall the proof of convergence of for p > 1. Ren for...
Solve number 3 with good
details. Please do not use other's work.
1. Solve the equation 2. Prove that the series 00 n 2 converges at all points of the unit circle ll-1, except -1 3. Show that there is no convergent power series f(z)- 2, such that f (z) 1 forz 1/2, 1/3, 1/4,... and f(0) 0 l<
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3 (10) Let = Re', z = re (0<r< R) be two complex numbers. Show the following identities hold: R2 2 OO = Re = 1 +2 C-z ΣΑ. R2 - 2rR cos (-0)r2 coS n(-e) n=1