Question

find series' sum if the series is convergent
2+2/3+2/9+2/27+.......+2/3^(n-1)+.....

Answer 1

This is the sum of a geometric series: a, ar, ar^2, ar^3 ,.... where a = 2 and r = 1/3

It is convergent. And hopefully you know the formula for the sum.

Answer 2

how do u know it converges james

Answer 3

i study the sum of n terms is a1-r^n/(1-r) but u write r^(n+1) isn't it r^n???

Answer 4

Oh, yes, r^n because the first term is just a. You're right. But the rest of the arguments stand.

Answer 5

The sum of n terms,\[s_n = a { 1 - r^n \over 1 -r}\]
and in the limit,
\[\lim_{n \rightarrow \infty} s_n = {a \over 1 - r}\]

Answer 6

Now an example, the sum 1 + 1/2 + 1/4 + 1/8 + 1/16 + ..... Here a = 1 and r = 1/2

Thus the infinite sum exists and is equal to
\[{ 1 \over 1 - 1/2} = {1 \over 1/2} = 2\]

Answer 7

i want to clear something
we take the limit just to see if the series convergent or not. if limit exist then we say that it converges if limit is infinity or not exist then it diverges right??

Answer 8

It's better logically that we first know it is convergent.

The mini-theorem here is this:
Let a, ar, ar^2, ar^3, ... be geometric series.
Then the infinite sum of the terms a + ar + ar^2 + ar^3 + .... exists
if and only if |r| < 1
and in that case that sum is equal to
a/(1-r)

Answer 9

However you can reprove this mini-theorem each time if you want, but the result should be considered standard after a while.

Answer 10

and james what do u say about 0^0.
that's the question confusing me sometimes

Answer 11

It depends on the context: sometimes we will say it is 1, sometimes we will say it is not defined. And in doing so it's not that mathematics is inconsistent, but depending on what definition of x^y you are using at given time, 0^0 will either exist or it will not.

Answer 12

but logically it should be undefined
i.e 0^0=0^1-1=0/0=undefined

Answer 13

Yes, the first definition of 0^0 should be that it is undefined.

However, later on in mathematics function such as
f(x) = Ax^0 + Bx^1 + Cx^2
arise as the natural formulation and working of problems such as ordinary differential equations and there it would be highly inconvenient to insist that this function is not defined at x = 0, so we gloss the issue and just say that A.0^0 = A.1 = A.

Answer 14

In high school, which is where I'm guessing you are, the safe assumption is that 0^0 is not defined.

Answer 15

another and last que is:
is e the anti-log of natural log?

Answer 16

f(x) = e^x is the inversion function of g(x) = ln(x)

Answer 17

In fact, in a careful university-level development of calculus, this is how e^x is usual defined.

Answer 18

thanks a lot