Proof of limit comparison test - sequences and series, Mathematics

Assignment Help:

Proof of Limit Comparison Test

As 0 < c <∞ we can find out two positive and finite numbers, m and M, like m < c < M .

 Now, as  

888_Proof of Limit Comparison Test - Sequences and Series.png

we know that for large enough n the quotient an/bn should be close to c and thus there must be a positive integer N like if n > N we as well have,

m < an / bn < M

Multiplying by bn provides

Mbn < an < Mbn

provided n > N .

Here now, if ∑bn diverges then thus does ∑mbn and so as mbn < an for all adequately large n by the Comparison Test ∑an as well diverges. 

Similarly, if  ∑bn converges then so does ∑Mbn and as an < Mbn for all sufficiently large n by the Comparison Test ∑an as converges.


Related Discussions:- Proof of limit comparison test - sequences and series

6, 200000+500

200000+500

Find out the absolute extrema for function and interval, Find out the absol...

Find out the absolute extrema for the given function and interval.  g (t ) = 2t 3 + 3t 2 -12t + 4 on [-4, 2] Solution : All we actually need to do here is follow the pr

People fit, How many people ca fi in a small cars without seats?

How many people ca fi in a small cars without seats?

What is the smaller probable value of this number, The sum of the square of...

The sum of the square of a number and 12 times the number is -27. What is the smaller probable value of this number? Let x = the number.  The statement that is "The sum of the

Prove gcd value, Let a, b, c 2 Z + . (a) Prove that if a|b, then ac|bc f...

Let a, b, c 2 Z + . (a) Prove that if a|b, then ac|bc for all c. (b) If a|bc, can you conclude that either a|b or a|c? Justify your answer with a proof or a counter example.

Express the product of -9p3r and the quantity 2p - 3r, Express the product ...

Express the product of -9p3r and the quantity 2p - 3r in simplified form. The translated expression would be -9p3r(2p - 3r). Noticed that the key word product means multiply.

Math, A small square is located inside a bigger square. The length of the s...

A small square is located inside a bigger square. The length of the small square is 3 in. The length of the large square is 7m. What is the area of the big square if you take out t

Partial Differential Equations Walter A Strauss, Find the full fourier Seri...

Find the full fourier Series of e^x on (-l,l)in its real and complex forms. (hint:it is convenient to find the complex form first)

Give an examples of simplifying fractions , Give an examples of Simplifying...

Give an examples of Simplifying Fractions ? When a fraction cannot be reduced any further, the fraction is in its simplest form. To reduce a fraction to its simplest form,

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd