Question

setbuilder form of 1,1/2,1/3,1/4,1/5,1/6

Answer 1

Answer:

Step-by-step explanation:

loge2=1−12+13−14+15−16...∞  

Proof:

First of all we need to prove that

ex=1+x+x22!+x33!+....∞

Or simply,

ex=∑k=0∞xkk!

Actually,

ex=limn→∞(1+n)xn=limn→∞(1+1n)nx

According to binomial expansion,

ex=limn→∞[1+nxn+n2!(n−1)(xn)2+n3!(n−1)(n−2)(xn)3...∞]

After simplifying (dividing by n),

ex=limn→∞[1+x+x22!(1−1n)+x33!(1−1n)(1−2n)...∞]

Applying limit,

ex=1+x+x22!+x33!...∞

Now replacing x with xlogea,

∴exlogea=1+xlogea+x22!(logea)2+x33!(logea)3...∞

∵xlogea=logeax

To prove:

nlogem=logemn

Proof:

Let,

nlogemlogemmmnk∴nlogem=k=kn=ekn=ek=logemn=logemn(Dividing by n)(Antilog on both sides)(Power n)(Log on both sides)(from equations 1 & 2)(1)(2)

∴elogeax=1+xlogea+x22!(logea)2+x33!(logea)3...∞

∵elogeax=ax

To prove:

alogam=m

Proof:

Let,

alogamlogamak∴alogam=ak=k=m=m(Power of a)(Antilog on both sides)(From equations 1 & 2)(1)(2)

∴ax=1+xlogea+x22!(logea)2+x33!(logea)3...∞

Now replacing a with (1+x) and x with n,

(1+x)n=1+nloge(1+x)+n22![loge(1+x)]2+n33![loge(1+x)]3...∞(1)

And by binomial expansion,

(1+x)n=1+nx+n2!(n−1)x2+n3!(n−1)(n−2)x3...∞(2)

From equations 1 & 2,

1+nloge(1+x)+n22![loge(1+x)]2+n33![loge(1+x)]3...∞=1+nx+n2!(n−1)x2+n3!(n−1)(n−2)x3...∞

On comparing coefficients of n

∴loge(1+x)=1−x+x22−x33+x44...∞

Now replacing x with 1,

∴loge2=1−12+13−14+15−16...∞

Hence proved

Thank you for reading my answer _/\_

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Frank Wei

Frank Wei, Loves math as he loves his girlfriend, but has no girlfriend

Answered Sep 10, 2017 · Author has 232 answers and 687.7k answer views

We can start off with the sum

∑k=12n(−1)k+1k

and take the limit as n tends towards infinity. Manipulating the series, we have

∑k=12n(−1)k+1k=1−12+13−14+…+12n−1−12n=∑k=12n1k−∑k=1n1k

Using the definition of the Euler - Mascheroni constant

limx→∞{∑k≤x1k−logx}=γ

We have the limit as

limn→∞{∑k=12n(−1)k+1k}=limn→∞{∑k=12n1k−∑k=1n1k}=limn→∞{∑k=12n1k−log2n}−limn→∞{∑k=1n1k−logn}+log2=log2

Therefore, the limit as it tends to infinity is log2.