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:
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.
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.
The n-th partial sum of the series is,
Solutions are written by subject experts who are available 24/7. Questions are typically answered within 1 hour.*See Solution
Q: Solve y ″ + 4 y ′ + 4 y = x e − 2 x + cos ( 2 x ) Correct! y = ( c 1 + c 2 x + 1 6 x 3 ) e − 2 x +...
A: Click to see the answer
Q: You want to be able to withdraw $40,000 each year for 25 years. Your account earns 5% interest.a) Ho...
A: Consider the given question,Given regular withdraw amount(d) = $40,000Number of years(n) = 25 yearsR...
Q: how do i calculate annuities
A: An annuity is a series of periodic payments that are received at a future date.The annuity payment f...
Q: Problem 7. (10 pts) Write a dual problem of the following primal problem and find the optimal soluti...
A: The given primal problem is
Q: Find the eigenvalues and eigenvectors of the following matrices and check your answer by computer 2 ...
A: Click to see the answer
Q: A store is hiring 5 new stockers for their evening shift. 10 equally qualified students apply for th...
A: Here we using the formula: The number of ways to select p people from a total of n people is:
Q: Find the standard form of the equation of the ellipse satisfying the given conditions. Foci: (-2,0),...
Q: In Exercises 15-19 find the general solution. 15. y3y +2y e3x(1 + x)
A: To find a (any) function y=y(x) satisfying the given non-homogeneous equation
Q: Hi, parts a) and b) of this question are attached. Thanks so much!!!
A: The problem concerns the distance and angle functions in the complex plane (the set of all complex n...