- #1

Edwin

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Suppose you start with the number 0. If you Integrate 0 you get 1.

Next, if you integrate 1 to get x + C, but for simplicity, let C =

0, so you get just x. If you integrate the result again, that is,

if you Integrate x, you get (x^2)/2 + C2, but for simplicity, let C2

also = 0, so you get just (x^2)/2. Let all arbitrary constants Cn,

of the given sequence of polynomial antiderivitives = 0 for

simplicity's sake. If you integrate each result an infinite number

of times, you would get an infinite sequence of antiderivatives of

the number 0.

From this you could create an infinite series from these anti-

derivitives as follows:

1+x + (x^2)/2 + (x^3)/6 + (x^4)/24...etc.

The question I have is, is the infinite series ever convergent?

If so, for what values of x is the sequence convergent.

If the sequence is convergent for some values of x, and x=1 happens

to be one of those values, then what does the sum of this infinite

series converge to

if x = 1? The answer may, or may not, surprise you!

Inquisitively,

Edwin G. Schasteen

Next, if you integrate 1 to get x + C, but for simplicity, let C =

0, so you get just x. If you integrate the result again, that is,

if you Integrate x, you get (x^2)/2 + C2, but for simplicity, let C2

also = 0, so you get just (x^2)/2. Let all arbitrary constants Cn,

of the given sequence of polynomial antiderivitives = 0 for

simplicity's sake. If you integrate each result an infinite number

of times, you would get an infinite sequence of antiderivatives of

the number 0.

From this you could create an infinite series from these anti-

derivitives as follows:

1+x + (x^2)/2 + (x^3)/6 + (x^4)/24...etc.

The question I have is, is the infinite series ever convergent?

If so, for what values of x is the sequence convergent.

If the sequence is convergent for some values of x, and x=1 happens

to be one of those values, then what does the sum of this infinite

series converge to

if x = 1? The answer may, or may not, surprise you!

Inquisitively,

Edwin G. Schasteen

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