# Question:How do I find the interval of convergence of a Taylor series?

## Question:How do I find the interval of convergence of a Taylor series?

Maple

I have to copy and paset c(n), and my final results are the square root of what they should be.

```

f:=(x-1)/(x+2)
-1 + x
f := ------
x + 2

plot(f,x=-3..3)

sum(k -> 1/k^2, k=1..1000)
1000 f

series(f,x=0);
1   3     3  2   3   3   3   4   3   5    / 6\
- - + - x - - x  + -- x  - -- x  + -- x  + O\x /
2   4     8      16      32      64

s:=convert(f,FormalPowerSeries)
/infinity           \
| -----             |
|  \     /      k  \|
|   )    |   3 x   ||
s := 1 + |  /     |- -------||
| -----  |        k||
\ k = 0  \  2 (-2) //

simplify(s,symbolic)
/infinity                       \
| -----                         |
|  \                            |
|   )    /       k  (-k - 1)  k\|
1 + |  /     \-3 (-1)  2         x /|
| -----                         |
\ k = 0                         /

c := c

c:=k->-(3 *x^k)/((2 (-2)^k))
c(k)
k
3 x
- ----
k
2  x

abs( ((c(k+1)*(x^(k+1)))) / ((c(k)*x^k)));
2 |    k   |
| (k + 1)|  |   2    |
|x       |  |--------|
| (k + 1)|
|2       |
----------------------
2
| k|
|x |

simplify( % );
2
| (k + 1)|
|x       |
-----------
2
| k|
2 |x |

L:= limit( %, k=infinity );
1    2
L := - |x|
2

solve( L<1, x);

```

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