A formula for approximating ln(2) and sums of other alternating series

In summary, the conversation discusses a mathematical inequality involving a series of fractions and the natural logarithm. The person has rediscovered and proved this inequality through induction and has found that it holds true for a specific value of n. They also mention that without one of the fractions, the inequality does not hold. The conversation then moves on to discussing series expansions and convergence properties, providing links for further information and suggesting the use of theorems to show convergence.
  • #1
checkitagain
138
1
[tex]1 \ - \ \frac{1}{2} \ + \ \frac{1}{3} \ - \ \frac{1}{4} \ + \ ... \ - \ \frac{1}{n - 1} \ + \ \frac{1}{n} \ - \ \frac{1}{2n + 1} \ < \ ln(n),[/tex] where n is a positive odd integer



I worked this out (rediscovered it) and proved it by induction.

For example, when n = 71 (summing of 71 terms and that 72nd fraction),

it gives five correct decimal digits (and six correct decimal digits

when rounded).


[tex]But, \ \ without \ the \ \frac{1}{2n + 1} \ term \ to \ be \ subtracted, \ \ it \ gives \ \ 0.7... \
(zero \ \ correct \ \ decimal \ \ digits).[/tex]
 
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  • #2
Hey checkitagain.

Have you seen the wikipedia page? If you want to know more about these expansions you need to understand what is known as a series expansion: the taylor series is a good start. Take a look at these links:

http://en.wikipedia.org/wiki/Natural_logarithm#Derivative
http://en.wikipedia.org/wiki/Taylor_series

In terms of convergence properties, that is a whole other kettle of fish. If you want to know about showing convergence for this kind of thing then there are a variety of theorems you can use.

This might give you a start:

http://en.wikipedia.org/wiki/Convergence_(mathematics)#Convergence_and_fixed_point
 

Related to A formula for approximating ln(2) and sums of other alternating series

1. What is the formula for approximating ln(2)?

The formula for approximating ln(2) is 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ...

2. How is this formula derived?

This formula is derived from the power series expansion of ln(x) at x = 1. By plugging in x = 2, we can approximate ln(2).

3. Can this formula be used to approximate other alternating series?

Yes, this formula can be used to approximate other alternating series by replacing the 1/n terms with the corresponding terms in the series. For example, to approximate ln(3), we would use 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ...

4. What is the accuracy of this formula?

The error in using this formula to approximate ln(2) decreases as more terms are added. After n terms, the error is less than 1/(n+1), meaning that for large n, the error becomes negligible.

5. How can this formula be used in real-world applications?

This formula can be used to quickly approximate ln(2) and other alternating series in various scientific and mathematical calculations. It can also be used as a tool for teaching and understanding the concept of power series expansions.

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