16. Given a sequence (an) of positive integers, we define1+a11and Rn =an11+11+11+AnSn =...a2Show that if (Sn) converges, then (In Rn) also converges.117. Let (Hn) be a sequence defined by HnΣk=11< In(n + 1) – In n < =.1(a) Show that for n > 0,n +1n(b) Deduce that In(n + 1) < Hn < Inn + 1Determine the limit of Hn.(d) Show that unHn - Inn is decreasing and positive.

Name: 16. Given a sequence (an) of positive integers, we define1+a11and Rn
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Description: 16. Given a sequence (an) of positive integers, we define1+a11and Rn =an11+11+11+AnSn =...a2Show that if (Sn) converges, then (In Rn) also converges.117. Let (Hn) be a sequence defined by HnΣk=11 0,n +1n(

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16. Given a sequence (an) of positive integers, we define 1 + a1 1 1 and Rn = + an 1 1+ a1 1 1+ an Sn 1+ a2 - a2 Show that if (Sn) converges, then (In Rn) also converges. 1 17. Let (Hn) be a sequence defined by Hn =)) k' k=1 1 (a) Show that for n > 0, 1 < In(n + 1) – Inn < n+1 n (b) Deduce that In(n + 1) < Hn < In n +1 (c) Determine the limit of Hn: (d) Show that Un = Hn – In n is decreasing and positive. %3D

16. Given a sequence (an) of positive integers, we define 1 + a1 1 1 and Rn = + an 1 1+ a1 1 1+ an Sn 1+ a2 - a2 Show that if (Sn) converges, then (In Rn) also converges. 1 17. Let (Hn) be a sequence defined by Hn =)) k' k=1 1 (a) Show that for n > 0, 1 < In(n + 1) – Inn < n+1 n (b) Deduce that In(n + 1) < Hn < In n +1 (c) Determine the limit of Hn: (d) Show that Un = Hn – In n is decreasing and positive. %3D

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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Concept explainers

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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16. Given a sequence (an) of positive integers, we define
1
+
a1
1
and Rn =
an
1
1+
1
1+
1
1+
An
Sn =
...
a2
Show that if (Sn) converges, then (In Rn) also converges.
1
17. Let (Hn) be a sequence defined by Hn
Σ
k=1
1
< In(n + 1) – In n < =.
1
(a) Show that for n > 0,
n +1
n
(b) Deduce that In(n + 1) < Hn < Inn + 1
Determine the limit of Hn.
(d) Show that un
Hn - Inn is decreasing and positive.

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