Posted by **jack** on Sunday, June 16, 2013 at 6:51pm.

estimate ln(e^2+0.1)-ln(e^2)

- calculus -
**Damon**, Sunday, June 16, 2013 at 7:44pm
did this below

- calculus -
**Count Iblis**, Sunday, June 16, 2013 at 7:46pm
Log(e^2 + 1/10) =

Log(e^2) + Log[1+e^(-2)/10]

We can compute this using the series expansion:

Log(1+x) = x - x^2/2 + x^3/3 - x^4/4 + ...

But we can speed up the convergence of the series as follows. Replacing x by

-x in the series gives:

Log(1-x) = -x - x^2/2 - x^3/3 - x^4/4 + ...

Subtract the two series to obtain:

Log[(1+x)/(1-x)] =

2[ x + x^3/3 + x^5/5 + x^7/7 +...]

If we want to compute log(1+u) for small u, we must choose x such that:

(1+x)/(1-x) = 1+u ------>

x = u/(2+u). So, we have:

Log[1+e^(-2)/10] =

2 e^(-2)/10 / (2 + e^(-2)/10 ) +

2/3 e^(-6)/10^3 / [(2 + e^(-2)/10 )]^3

+ 2/5 e^(-10)/10^5 / [(2 + e^(-2)/10)]^5

+...

The first 3 terms will give the answer to an accuracy of 10^(-16).

## Answer this Question

## Related Questions

- Calculus - calculate the first 30 partial sums for: A)1- 1/2 + 1/3 - 1/4 + 1/5...
- calculus III - calculate the first 30 partial sums for: A)1- 1/2 + 1/3 - 1/4 + 1...
- Calculus - t 2 4 6 8 10 12 v(t) 44 42 41 40 37 35 the velocity in the table is ...
- calculus - f(1)= 20, f(3)=13, f(5)=15, f(7)=16, f(9)=11, on [0,6] a, used ...
- calculus - Estimate 1/(sqrt(82)) using linear approximation and the exact value...
- ALGEBra Estimate - need to estimate to find quotient: -476 / 90 ESTIMATE IS -450...
- Calculus - Find the local linearization of g(x)=sqrt 4x near x=2 use local ...
- calculus - For time, t, in hours, 0\le t\le1, a bug is crawling at a velocity, v...
- Calculus - Use the table of values to estimate ⌠6 ⎮ f(x)⋅dx &#...
- Calculus - estimate the absolute maximum and minimum values of f if any, on the ...

More Related Questions