. Let (xn)neN be a sequence in R. For each n in N, let X1 +x2 + +Xn Yn Show that if (xm)nɛN converges to x, then (yn)neN converges to x. [Hint: Write x1 +x2 + · ··+Xn nx Yn -x = (x- Oux) +.+ (x-Ix) (xno+1 - x) + + (xn – x) n and, given ɛ > 0 and suitably choosing no, |x1 - x++xno - x| п - по E. ... lyn – x| < Now take the limit superior of both sides of this inequality.] 5. Refer to Exercise 5. Show that there are nonconvergent sequences Xn)neN for which (yn)neN Converges. [Hint: Consider (0, 1,0, 1, 0, 1, ...).]
. Let (xn)neN be a sequence in R. For each n in N, let X1 +x2 + +Xn Yn Show that if (xm)nɛN converges to x, then (yn)neN converges to x. [Hint: Write x1 +x2 + · ··+Xn nx Yn -x = (x- Oux) +.+ (x-Ix) (xno+1 - x) + + (xn – x) n and, given ɛ > 0 and suitably choosing no, |x1 - x++xno - x| п - по E. ... lyn – x| < Now take the limit superior of both sides of this inequality.] 5. Refer to Exercise 5. Show that there are nonconvergent sequences Xn)neN for which (yn)neN Converges. [Hint: Consider (0, 1,0, 1, 0, 1, ...).]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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QUESTIONS 5 AND 6 PLEASE
QUESTION 6 RELIES ON QUESTION 5 TO GET THE ANSWER
![5. Let (xn)neN be a sequence in R. For each n in N, let
x1 +x2 + .+ Xn
...
Yn =
n
Show that if (xn)neN converges to x, then (yn)neN converges to x. [Hint:
Write
x1 +x2+··+ Xn
nx
Yn - x =
(xno+1 - x) + + (xn - x)
(x- Oux) + + (x-Ix)
...
|
n
and, given ɛ > 0 and suitably choosing no,
|x1 - x++ |Xno-x
п - по
E.
..
lyn – x| <
Now take the limit superior of both sides of this inequality.]
6. Refer to Exercise 5. Show that there are nonconvergent sequences (Xn)neN
for which (y,)neN converges. [Hint: Consider (0, 1, 0, 1, 0, 1, ...).]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd95b1c5e-3673-40a0-a87b-179a3e73672c%2F5ac2cb0d-08c8-4c49-9eb7-50549a5b9514%2Flbq1e9_processed.png&w=3840&q=75)
Transcribed Image Text:5. Let (xn)neN be a sequence in R. For each n in N, let
x1 +x2 + .+ Xn
...
Yn =
n
Show that if (xn)neN converges to x, then (yn)neN converges to x. [Hint:
Write
x1 +x2+··+ Xn
nx
Yn - x =
(xno+1 - x) + + (xn - x)
(x- Oux) + + (x-Ix)
...
|
n
and, given ɛ > 0 and suitably choosing no,
|x1 - x++ |Xno-x
п - по
E.
..
lyn – x| <
Now take the limit superior of both sides of this inequality.]
6. Refer to Exercise 5. Show that there are nonconvergent sequences (Xn)neN
for which (y,)neN converges. [Hint: Consider (0, 1, 0, 1, 0, 1, ...).]
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