part e and f 0 for all k E N and Σ at oo. For each of the following, either prove that the given series con- 4. Suppose ak verges, or provide an example for which the series diverges. ak 1 + at ar ai ak 0 for all k E N and Σ at oo. For each of the following, either prove that the given series con- 4. Suppose ak verges, or provide an example for which the series...
lf {Ζ., n>0) is a martingale, is this true for E[Zw | Z.]= Zn for n>0? Please prove. d lf {Ζ., n>0) is a martingale, is this true for E[Zw | Z.]= Zn for n>0? Please prove. d
3.) Let ak E R with ak > 0 for all k E N. Suppose Σ㎞iak converges. Show that Σί1bk (By definition, for a sequence (ck), we say liCkoo if, for all M ER with Hint: Show that there exists (Ni))ไ1 with N > Nj for all j E N, such that bk there exists a sequence (bk)k of real numbers such that lim converges = oo and M >0, there exists N E N such that ck > M...
13 please 8. b. -2 3 0 0 0 0 -1 2 0 0-4 0 3 0-2 0 3 0 0 -2 0 3 0 4 o0-1 6 0 0 1 o 2 6 0 0 -1 6 10. For any positive integer k, prove that det(4t) - de(A)*. 11. Prove that if A is invertible, then den(A-1)- I/der(A) - det(4)- 12. We know in general that A-B丰B-A for two n x n matrices. However, prove that: det(A . B)-det(B...
Soru 2. Suppose the series Ebox" converges for (x1<4. Select all that applies n=0 Yanıtınız: (-1)"b,4" converges n=0 6" M8 M 0,4" converges. n=0 61 6,6"4" diverges. n=0 00 (-1)"b,4" diverges. n=0 boxn+1 n=0 n +1 È converges for all |x<4 nb bmxn-1 converges for |x|<4 n=1 Testi duraklat Geri Sonraki
(14.3) Suppose that f()-OP0cman for z E C. Prove that, for all R. where ) n=0 (14.3) Suppose that f()-OP0cman for z E C. Prove that, for all R. where ) n=0
Suppose that an >0 and bn >0 for all n2N (N an integer). If lim = , what can you conclude about the convergence of an? A. a, diverges if by diverges, and an converges if bn converges. an diverges if by diverges. c. a, converges if be converges. OD. The convergence of an cannot be determined.
For all n E N prove that 0 <e- > < 2 k!“ (n + 1)! k=0 Hint: Think about Taylor approximations of the function e".
Need to use all axioms to prove this is a vector space. e(a+b)z and scalar multiplication as feax a E R} define addition as ea* + ebx ekax where k e R. Is V a vector space under these definitions? If so, what is the 0 element = eaeba- 8. Let V = k ea of V? e(a+b)z and scalar multiplication as feax a E R} define addition as ea* + ebx ekax where k e R. Is V a...
00 and an+12a, >0 for all n21. Which of the following ste ing statements 3. Suppose lima, be true? M. Α. Σ. diverges. B. § (-1)", converges. c. & converges. D. Šal) converges. E. Ea, converges. M IM IM. requires This world justification) 4. Which of the following series is conditionally convergent? KH C D which Sha E ŽGO