3. Assume the sequence of positive numbers (sn) increases monotonicallyto infinity. Prove thatSn+1SnSn+1Sn--Σ,Σ< 0.Sns2'n+1Snn

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Description: 3. Assume the sequence of positive numbers (sn) increases monotonicallyto infinity. Prove thatSn+1SnSn+1Sn--Σ,Σ< 0.Sns2'n+1Snn

Answered: 3. Assume the sequence of positive…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.1: Infinite Sequences And Summation Notation

Problem 74E

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3. Assume the sequence of positive numbers (sn) increases monotonically
to infinity. Prove that
Sn+1
Sn
Sn+1
Sn
-
-
Σ
,
Σ
< 0.
Sns2
'n+1
Sn
n

Expert Solution

Step 1

Let $S_{n}$ be a sequence of positive numbers $(s_{n})$ increase monotonically to infinity.

$\Rightarrow \lim_{n \to \infty} s_{n} = \infty (∵ s_{n} i s m o n o t o n i c a l l y i n c r e a \sin g t o i n f i n i t y)$

Now,

$\sum_{n = 1}^{\infty} \frac{s_{n + 1} - s_{n}}{s_{n}} = \sum_{n = 1}^{\infty} (\frac{s_{n + 1}}{s_{n}} - \frac{s_{n}}{s_{n}}) = \sum_{n = 1}^{\infty} (\frac{s_{n + 1}}{s_{n}} - 1)$

By $n t h t e r m t e s t$

$\lim_{n \to \infty} (\frac{s_{n + 1}}{s_{n}} - 1) = \lim_{n \to \infty} (\frac{s_{n + 1}}{s_{n}}) - \lim_{n \to \infty} 1 = a - 1 (w h e r e a > 1) \neq 0 (∵ i f <a_{n}> i s a p o s i t i v e t e r m s e q u e n c e a n d i s d i v e r g e n t t h e n \lim_{n \to \infty} \frac{a_{n + 1}}{a_{n}} > 1)$

Step 2

since nth term of this series is non zero so, it is not converging series.

$i . e$ $\sum_{n = 1}^{\infty} \frac{s_{n + 1} - s_{n}}{s_{n}}$ is not converging series.

since $(s_{n})$ is sequence of positive terms and monotonically increasing sequence therefore $<\frac{s_{n + 1} - s_{n}}{s_{n}}>$ is a positive term sequence of non repeated numbers.

$\Rightarrow \sum_{n = 1}^{\infty} \frac{s_{n + 1} - s_{n}}{s_{n}}$ is divergent and diverge to $+ \infty$

hence $\sum_{n = 1}^{\infty} \frac{s_{n + 1} - s_{n}}{s_{n}} = \infty$

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3. Assume the sequence of positive numbers (sn) increases monotonically to infinity. Prove that Sn+1 Sn Sn+1 Sn Σ Σ <∞. || Sns2 n+1 Sn n n