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Advanced Calculus Homework 7: Limits of Sequences - Prof. Wu Jing, Assignments of Advanced Calculus

The fifth homework assignment for the advanced calculus course (math 412) at a university. It includes five problems related to the limits of sequences, such as showing that the limit of a sequence plus a constant is equal to the sum of the limit and the constant, and that the limit of a non-negative sequence is less than or equal to zero if all terms are non-positive.

Typology: Assignments

Pre 2010

Uploaded on 08/01/2009

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koofers-user-wsf 🇺🇸

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MATH 412 ADVANCED CALCULUS HOMEWORK #7
Due November 7, 2007
1. Let X= (xn)Rbe a sequence such that lim(xn) = x, and aR. Use the
K() definition of the limit a sequence to show that lim(xn+a) = x+a.
2. Let X= (xn)Rbe a sequence such that lim(xn) = xand xn0 for all nN.
Show that x0.
3. Let X= (xn)Rbe a sequence. Show that lim(xn) = 0 if and only if lim(x2
n) = 0.
(Hint: Use x2
n=|xn|2)
4. Let X= (xn), Y = (yn)Rbe two seuqences with lim xn= 0. Show that
lim(xnsin(yn)) = 0.
5. Let X= (xn)Rbe a sequence such that lim(xn) = xand x > 0. Show that
there exists KNsuch that xn>0 for all n>K. (Hint: Choose =x
2)
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MATH 412 ADVANCED CALCULUS HOMEWORK

Due November 7, 2007

  1. Let X = (xn) ⊆ R be a sequence such that lim(xn) = x, and a ∈ R. Use the  − K() definition of the limit a sequence to show that lim(xn + a) = x + a.
  2. Let X = (xn) ⊆ R be a sequence such that lim(xn) = x and xn ≤ 0 for all n ∈ N. Show that x ≤ 0.
  3. Let X = (xn) ⊆ R be a sequence. Show that lim(xn) = 0 if and only if lim(x^2 n) = 0. (Hint: Use x^2 n = |xn|^2 )
  4. Let X = (xn), Y = (yn) ⊆ R be two seuqences with lim xn = 0. Show that lim(xn sin(yn)) = 0.
  5. Let X = (xn) ⊆ R be a sequence such that lim(xn) = x and x > 0. Show that there exists K ∈ N such that xn > 0 for all n > K. (Hint: Choose  = x 2 )