Question

The series `sum_(n=1)^oo x^n/10^n` converges for `|x| lt beta`, find `beta`?

Answer 1

`beta=10`

Explanation:

We can apply d'Alembert's ratio test:

Suppose that;

`S=sum_(n=1)^oo a_n \ \ `, and `\ \ L=lim_(n rarr oo) |a_(n+1)/a_n|`

Then

if L < 1 then the series converges absolutely;
if L > 1 then the series is divergent;
if L = 1 or the limit fails to exist the test is inconclusive.

Our series is;

`S = sum_(n=1)^oo a_n` with `a_n=x^n/10^n`

So our test limit is:

`L = lim_(n rarr oo) | ( x^(n+1)/10^(n+1) ) / ( x^n/10^n) |`
`\ \ \ = lim_(n rarr oo) | ( x^(n+1)/10^(n+1) ) * ( 10^n/x^n ) |`
`\ \ \ = lim_(n rarr oo) | ( (x x^n)/(10 * 10^n) ) * ( 10^n/x^n ) |`
`\ \ \ = lim_(n rarr oo) | x/10 |`
`\ \ \ = | x/10 |`

Comparing with the definition of the question, we can conclude that the series converges if `L lt 1`

`| x/10 | lt 1 => |x| lt 10`

Similarly, the series diverges if `L gt 1`

`| x/10 | gt 1 => |x| gt 10`

Hence, `beta=10`