Real AnalysisI posted this question earlier and got a response . I have further questions on the response.  I am including the URL of the previous response:https://www.bartleby.com/questions-and-answers/real-analysis-does-the-series-log11n-from-n1-to-infinity-converge-or-diverge-prove-the-convergence-o/97a36976-4bf8-4a81-bf97-1b9c2d442972My original question was:Prove that the series Sum of log(1+1/n) from n=1 to infinity converges or diverges.The questions I have on the response are as follows:1.  Why can I just change from log to ln? I seem to recall something about ln and log being interchangeable despite that log is log base 10 and ln is log base e.2.  Why does the sequence of partial sums become (ln(2)-ln(1))+(ln(3)-ln(2))+(ln(n)-ln(n-1))+(ln(n+1)-ln(n))=ln(n+1).I need a lot more detail.  I seem to be incredibly dense with Real Analysis, despite earning top grades in my previous classes.  There are always parts of the proofs that I do not understand which prevents me from understanding the material.  Thank you.

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Asked Oct 29, 2019
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Real Analysis

I posted this question earlier and got a response . I have further questions on the response.  I am including the URL of the previous response:

https://www.bartleby.com/questions-and-answers/real-analysis-does-the-series-log11n-from-n1-to-infinity-converge-or-diverge-prove-the-convergence-o/97a36976-4bf8-4a81-bf97-1b9c2d442972

My original question was:

Prove that the series Sum of log(1+1/n) from n=1 to infinity converges or diverges.

The questions I have on the response are as follows:

1.  Why can I just change from log to ln? I seem to recall something about ln and log being interchangeable despite that log is log base 10 and ln is log base e.

2.  Why does the sequence of partial sums become (ln(2)-ln(1))+(ln(3)-ln(2))+(ln(n)-ln(n-1))+(ln(n+1)-ln(n))=ln(n+1).

I need a lot more detail.  I seem to be incredibly dense with Real Analysis, despite earning top grades in my previous classes.  There are always parts of the proofs that I do not understand which prevents me from understanding the material.  Thank you. 

 

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Expert Answer

Step 1

1.

The general logarithm form is, loga x.

Suppose the base a = e , the logarithm form is, loge x which is called as natural logarithm.

Usually, the natural logarithm can be written as ‘ln x’.

Suppose the base a = 10 , the logarithm form is, log10 x which is called as common logarithm.

Usually, the common logarithm can be written as ‘log x’.

Clearly, log x and ln x are not same.

So, keep the sequence as it is.

Step 2

2.

The n-th partial sum of the series is,

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S log1+- k n Σlog! k log log,,a-log,b Elog (k1)log kwhere m,a,b > 0 and m 1 k-1 loga-logb log b

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Step 3

Known resu...

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Telescoping sums are finite sums in whi ch pairs of consecutive terms can cel each other, leaving only the initial and final terms a-a)=a-a0 or (a*-a,)= a- a k1 k1

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