#MathsInMinutes Day 48: Series and approximations

Before we start with today´s entry I would like to heartfeltly thank @Ottman, who has very generously donated 0 Tokens to support this ongoing series. These 0 Tokens will be used to boost the post in hopes of reaching a wider audience.  

------------------------------------------------------------------------------------------------------------------------

#MathsInMinutes Day 48: Series and approximations

Some of the fundamental numbers of mathematics arise as infinite sums, and so these series can be used to find approximations to numbers, such as pi, e and some natural logarithms.

The harmonic series, 1+1/2+1/3+1/4+1/5+..., is a good place to start. By changing every other plus sign to a minus, the sum converges on the value of the natural logarithm of 2. And by replacing the denominator of each fraction with its square, the sum converges on the number pi²/6. In fact, every sum of even powers converges on a known constant multiplied by a power of pi². The sums of odd powers also converge, but to numbers without a known closed-form expression.

Finally, if we replace each denominator with its factorial, the sum converges on e. A factorial, represented by the symbol !, is the product of a number multiplied by all the positive numbers below it. So 3! = 3x2x1 = 6 and 5! = 5x4x3x2x1 = 120.

------------------------------------------------------------------------------------------------------------------------

I hope you enjoyed!

If you liked this content consider buying the original book:

http://a.co/72sJoVV

Or donating any amount of Tokens to help me boost these entries.

If you want to be notified about a new blog entry in this series, please leave a short message below along the lines of "i am interested" so I can write your handle down and @ you in the next entry.