4. For a sequence {n}, we define the arithmetic mean sequence {n} as X1 + X2+...+Xn n We will see how the convergence of {n} relates to the convergence of {x}. Note this question does not require any knowledge of series. (a) Let {n} = {(-1)"}. Show that {n} converges. (b) Suppose lim n = L. Let > 0 be arbitrary, and take M N such that for all n> M, xn-L| < E/2. Show that there is some K 20 such that for all n > M, we have MK (n-M)e 2n n |- L≤ + (Hint: Use In-L=E=1 (c) Note the right hand side of the inequality in (b) converges to e/2 as n → ∞. Prove that there exists some M'EN such that for all n > M', with the triangle inequality.) MK (n-M)e 2n + n and use it to conclude that if lim x₁ = L, then lim ₁ = L.
4. For a sequence {n}, we define the arithmetic mean sequence {n} as X1 + X2+...+Xn n We will see how the convergence of {n} relates to the convergence of {x}. Note this question does not require any knowledge of series. (a) Let {n} = {(-1)"}. Show that {n} converges. (b) Suppose lim n = L. Let > 0 be arbitrary, and take M N such that for all n> M, xn-L| < E/2. Show that there is some K 20 such that for all n > M, we have MK (n-M)e 2n n |- L≤ + (Hint: Use In-L=E=1 (c) Note the right hand side of the inequality in (b) converges to e/2 as n → ∞. Prove that there exists some M'EN such that for all n > M', with the triangle inequality.) MK (n-M)e 2n + n and use it to conclude that if lim x₁ = L, then lim ₁ = L.
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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Transcribed Image Text:4. For a sequence {n}, we define the arithmetic mean sequence {n} as
¸X1 + X2 + ... + Xn
n
In
We will see how the convergence of {n} relates to the convergence of {n}. Note this
question does not require any knowledge of series.
(a) Let {n} = {(-1)"}. Show that {n} converges.
(b) Suppose lim x₁ = L. Let ɛ > 0 be arbitrary, and take M N such that for all
11-0
n ≥ M, |ïn − L\ < €/2.
Show that there is some K≥ 0 such that for all n ≥ M, we have
|ên-L|< +
(Hint: Use în - L = E=1
12
MK (n - M)e
2n
with the triangle inequality.)
MK
n
n
(c) Note the right hand side of the inequality in (b) converges to ɛ/2 as n → ∞.
Prove that there exists some M' EN such that for all n ≥ M',
+
(n - M)€
2n
and use it to conclude that if lim x₁ = L, then lim n = L.
11-x
004-11
Remark: As it turns out, not all bounded sequences have convergent mean sequences.
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