In search of pattern.

we know that ,
e^x = 1 + x/1! + x^2/2! + x^3/3! +...............ad. inf.

to get the series for a^x where a is any real number,
let e^y = a
thus log(a) = y
therefore a^x = e^(xy)

a^x = 1 + xy/1! + (xy)^2/2! + (xy)^3/3! +......................ad. inf.
a^x = 1 + x[log(a)]/1! +x[log(a)]^2/2! +x[log(a)]^3/3! +...............ad.inf.

now consider a = 1+y
we get,

(1+y)^x = 1 + x[log(1+y)]/1! +x^2.[log(1+y)]^2/2! +x^3.[log(1+y)]^3/3! +...............ad.inf. ---------(1)

also by binomial theorem,
(1+y)^x = 1 + x.y/1! + x.(x-1).y^2/2! + x.(x-1).(x-2).y^3/3! +.....................ad. inf. ----------(2)

from (1) & (2) , it is clear that log(1+y) is the coefficient of x in (2)
thus , log(1+y) = y - y^2/2 + y^3/3 + y^4/4 +..............ad. inf.

but thats not it , we also get to know that [log(1+y)]^2, [log(1+y)]^3, [log(1+y)]^4, and so on are the coefficients of x^2, x^3, x^4 and so on respectively in (2)

thats the thing everyone knows but still no book seem to have those infinite series ,
so i decided to calculate them and try to find some patterns in their coefficients.

log(1+y) = 1 + y/ 1 + y^2/2 + y^3/3 +................ad. inf. (you will find this series in most of the books)

what about raising an infinite series to higher integer power,

[log(1+y)]^2 = 2.[y^2/2! - 3.y^3/3! +11.y^4/4! - 50.y^5/5! + 274.y^6/6! - 1764.y^7/7! + 13068.y^8/8! - 109584.y^9/9! + 1026576.y^10/10! -............ad.inf]

[log(1+y)]^3 = 6.[y^3/3! - 6.y^4! + 35.y^5/5! - 225.y^6/6! + 1624.y^7/7! - 13132.y^8/8! +118124.y^9/9! - 1172700.y^10/10! + ................ad.inf.]

(this is special , you will not find this anywhere but here.

I was also calculating the series for [log(1+y)]^4 , but later i thought this is pointless, so i drop the idea (i was just hoping that maybe i will figure out some interesting things, but anyways i've done this just because i was curious.)


It is easy to see the pattern in the coefficient of the terms in log(1+y) series.
what about [log(1+y)]^2 and [log(1+y)]^3 ?
well i figured this out ,

the coefficient of y^n in the expansion of [log(1+y)]^2 is

2(-1)^n-1/n *(1 + 1/2 + 1/3 + 1/4 +.........1/n-1)

and for [log(1+y)]^3, nth term coeffcient is

(-1)^n+1 *[1 + 1/2 + 1/2^2 + ............ + 1/2^n-2]