![6. Suppose that {x,] is a sequence of positive numbers and limA = a Show that if L> 1 then lim x =00, and if L < 1 lim x = 0](http://img.homeworklib.com/questions/f05c3b70-cd32-11ea-986c-07657ac60168.png?x-oss-process=image/resize,w_560)
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6. Suppose that {x,] is a sequence of positive numbers and limA = a Show that...
1. Let {n} be a sequence of non negative real numbers, and suppose that limnan =...
1. Let {n} be a sequence of non negative real numbers, and suppose that limnan = 0 and 11 + x2 + ... + In <oo. lim sup - n-00 Prove that the sequence x + x + ... + converges and determine its limit. Hint: Start by trying to determine lim supno Yn. What can you say about lim infn- Yn? 3 ) for all n Expanded Hint: First, show that given any e > 0 we have (...
Problem 2 Show that if the sequence of numbers (an)n-1 satisfies Inlan) < oo, then the...
Problem 2 Show that if the sequence of numbers (an)n-1 satisfies Inlan) < oo, then the series In ancos(nx) converges uniformly on [0, 27). This means, the partial sums Sn(x) = ) ancos(nx) define a sequence of functions {sn} = that converges uniformly on [0, 271]. Hint: First show that the sequence is Cauchy with respect to || · ||00.
1,2 Let (an)nen be a sequence of real numbers that is bounded from above. Consider L...
1,2
Let (an)nen be a sequence of real numbers that is bounded from above. Consider L := lim suPn7o An, prove that: For all e > 0 there are only finitely many n for which an > L + €. For all e > 0 there are infinitely many n for which an > L - €.
real analysis. questions Prove that if lima In = 0 and > M for some M...
real analysis. questions
Prove that if lima In = 0 and > M for some M >0 and in 10 > 0, then lima (ny) - Asume 30 = 2,2-20+ In+1 = In + Prove that this sequence has a limit and find the limit. Prove that lim = L with L < if and only if every subsequence limo n L. Suppose that the sequence {an) is increasing and the sequence {yn) is decreasing. Moreover, lim a n -...
(10pts) 3. Use direct proof to show that if x and y are positive real numbers,...
(10pts) 3. Use direct proof to show that if x and y are positive real numbers, then (x+y)" > " + y".
Problem 6) (The Cauchy condensation test] Let {an} be a nonincreasing sequence of positive numbers (an...
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.
real analysis problem 1. sequence and series 2. 3. prove that please show me detail (for...
real analysis problem
1. sequence and series
2.
3. prove that
please show me detail (for beginner)
please don't use hand writing. please use typing
when. lima, –2. owe that lim (A -> ] when, lima, = 2, solve that lim (1-x) when f (x) = - n=in (a) show that given series are uniformly convergence in R (-00,00), (b) prove that f is uniformly continuous function in R (-00,00) prove with Taylor series (a) Σ = 6 (6) ΣΕΙ"...
an diverges. positive and decreasing sequence such that +ain- 39. Assume that (an) is a Show...
an diverges. positive and decreasing sequence such that +ain- 39. Assume that (an) is a Show that 1. lim + a2n n-oo
an diverges. positive and decreasing sequence such that +ain- 39. Assume that (an) is a Show that 1. lim + a2n n-oo
Suppose that A is diagonalizable and all eigenvalues of A are positive real numbers. Prove that...
Suppose that A is diagonalizable and all eigenvalues of A are
positive real numbers. Prove that det (A) > 0.
(1 point) Suppose that A is diagonalizable and all eigenvalues of A are positive real numbers. Prove that det(A) > 0. Proof: , where the diagonal entries of the diagonal matrix D are Because A is diagonalizable, there is an invertible matrix P such that eigenvalues 11, 12,...,n of A. Since = det(A), and 11 > 0,..., n > 0,...
HW: Show that the series __, an n=0 converges whenever ſal < 1, and diverges whenever...
HW: Show that the series __, an n=0 converges whenever ſal < 1, and diverges whenever al > 0.
1. Let {n} be a sequence of non negative real numbers, and suppose that limnan = 0 and 11 + x2 + ... + In <oo. lim sup - n-00 Prove that the sequence x + x + ... + converges and determine its limit. Hint: Start by trying to determine lim supno Yn. What can you say about lim infn- Yn? 3 ) for all n Expanded Hint: First, show that given any e > 0 we have (...
Problem 2 Show that if the sequence of numbers (an)n-1 satisfies Inlan) < oo, then the series In ancos(nx) converges uniformly on [0, 27). This means, the partial sums Sn(x) = ) ancos(nx) define a sequence of functions {sn} = that converges uniformly on [0, 271]. Hint: First show that the sequence is Cauchy with respect to || · ||00.
1,2
Let (an)nen be a sequence of real numbers that is bounded from above. Consider L := lim suPn7o An, prove that: For all e > 0 there are only finitely many n for which an > L + €. For all e > 0 there are infinitely many n for which an > L - €.
real analysis. questions
Prove that if lima In = 0 and > M for some M >0 and in 10 > 0, then lima (ny) - Asume 30 = 2,2-20+ In+1 = In + Prove that this sequence has a limit and find the limit. Prove that lim = L with L < if and only if every subsequence limo n L. Suppose that the sequence {an) is increasing and the sequence {yn) is decreasing. Moreover, lim a n -...
(10pts) 3. Use direct proof to show that if x and y are positive real numbers, then (x+y)" > " + y".
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.
real analysis problem
1. sequence and series
2.
3. prove that
please show me detail (for beginner)
please don't use hand writing. please use typing
when. lima, –2. owe that lim (A -> ] when, lima, = 2, solve that lim (1-x) when f (x) = - n=in (a) show that given series are uniformly convergence in R (-00,00), (b) prove that f is uniformly continuous function in R (-00,00) prove with Taylor series (a) Σ = 6 (6) ΣΕΙ"...
an diverges. positive and decreasing sequence such that +ain- 39. Assume that (an) is a Show that 1. lim + a2n n-oo
an diverges. positive and decreasing sequence such that +ain- 39. Assume that (an) is a Show that 1. lim + a2n n-oo
Suppose that A is diagonalizable and all eigenvalues of A are
positive real numbers. Prove that det (A) > 0.
(1 point) Suppose that A is diagonalizable and all eigenvalues of A are positive real numbers. Prove that det(A) > 0. Proof: , where the diagonal entries of the diagonal matrix D are Because A is diagonalizable, there is an invertible matrix P such that eigenvalues 11, 12,...,n of A. Since = det(A), and 11 > 0,..., n > 0,...
HW: Show that the series __, an n=0 converges whenever ſal < 1, and diverges whenever al > 0.
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