# series expansion of ln(x)

by jbowers9
Tags: expansion, series
 P: 70 How do you go about deriving the series expansion of ln(x)? 0 < x I got the representation at math.com but i'd still like to know how they got it. It's been a while since i did calc. iii. Thanks. John
 HW Helper P: 1,391 You expand it as you would any other function, you just have to be careful which point you expand it about. Typically the function is expanded about the point x = 1. The series expansion is $$\sum_{n=0}^{\infty}\frac{f^{(n)}(x_0)}{n!}(x - x_0)^n$$, so for $x_0 = 1$ and f(x) = lnx, the first few terms are $$\ln x_0 + \frac{1}{x_0}(x - x_0) - \frac{1}{2}\frac{1}{x_0^2}(x - x_0)^2 + ...$$ $$= (x - 1) - \frac{1}{2}(x - 1)^2 + ...$$ This can be made to look a little cleaner by letting u = x - 1.
 P: 70 I found the following link at math.com http://www.math.com/tables/expansion/log.htm I derived the first expression in the link and the one you gave the following way: 1/1-x = 1 + x + x^2 + x^3 + ... let x = -x and integrate 1/1+x = 1 - x + x^2 - x^3 int {1/1+x dx} = x - x^2/2 + x^3/3 - x^4/4 ... = ln(1+x) let x = x-1 ln(x) = (x-1) - (x-1)^2/2 + (x-1)^3/3 ... 0 < |x| < 1 The above expression is invalid at x=0. How are the three expressions following the first in the link derived?

 Related Discussions Calculus & Beyond Homework 1 Calculus & Beyond Homework 3 Calculus 4 Calculus & Beyond Homework 2 Calculus 2
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "http://www.physicsforums.com/showthread.php?t=203300", "fetch_time": 1394580332000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2014-10/segments/1394011352596/warc/CC-MAIN-20140305092232-00030-ip-10-183-142-35.ec2.internal.warc.gz", "warc_record_offset": 480884278, "warc_record_length": 6745, "token_count": 495, "char_count": 1410, "score": 4.25, "int_score": 4, "crawl": "CC-MAIN-2014-10", "snapshot_type": "longest", "language": "en", "language_score": 0.898189127445221, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00024-of-00064.parquet"}