Proof of limit comparison test - sequences and series, Mathematics

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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.


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