Does (1+x)^(-1) converge for |x|<1?

[Miike012]

In my book it says that the series (1+x)^n converges for x<1.

However I put n = -1 and wolfram says that the series does not converge.

However if I let x = 1/y where y>1

then the expansion of (1+1/y)^-1 is equal to: (which I will define as (SERIES 1))
1 - y + (1/y)² - (1/y)³ + (1/y)⁴ - ...

= 1 + ( (1/y)³ + (1/y)⁵ + ... ) - ( (1/y)² + (1/y)⁴ + ...)

The series (1/y)³ + (1/y)⁵ + ... (1/y)^{3 + 2n} + .. is equal to (which I will define as (SERIES 2))
Ʃ(1/y)^{3 + 2n} where n→∞ and 1≤n<∞.
I also know that the sum of the (SERIES 2) is less than the series 1 + 1/4 + 1/8 + 1/16 + ... which is convergent therefore (SERIES 2) is convergent.

Also from (SERIES 1) I know that the sum is positive and therefore the series
( (1/y)² + (1/y)⁴ + ...) is less than (SERIES 2) + 1 and therefore as the number of terms approaches ∞ the series
( (1/y)² + (1/y)⁴ + ...) which is positive is less than a [(finite number) + 1] which is a finite number and therefore is convergent.

∴Therefore the (SERIES 1) is convergent.


Am I and the book wrong or is wolfram?

 

[mfb]

In my book it says that the series (1+x)^n converges for x<1.

(1+x)^n is not a series.

If you want to sum that over natural n, it is pointless to set n to anything special (in particular, negative values).

Do you want to get the taylor expansion at x=0? That converges for x<1 for all n, and for all x for integer n (including 0 if we define 0⁰=1).

1 - y + (1/y)² - (1/y)³ + (1/y)⁴ - ...

= 1 + ( (1/y)³ + (1/y)⁵ + ... ) - ( (1/y)² + (1/y)⁴ + ...)

That step requires absolute convergence, so
1 + |1/y| + |(1/y)²| + |(1/y)³| ...
has to converge, too.

I also know that the sum of the (SERIES 2) is less than the series 1 + 1/4 + 1/8 + 1/16 + ... which is convergent therefore (SERIES 2) is convergent.

That is true for some y only.

Am I and the book wrong or is wolfram?

I would expect that the book and WolframAlpha are right. What did you use as query for WA?