What is Partial Sum of Power Series

Question

Consider the function1/(1−x^3)

Write a partial sum for the power series which represents this function consisting of the first 5 nonzero terms. For example, if the series were ∑n=0∞3n x^2n, you would write 1+3x^2+3^(2)x4+3^(3)x^6+3^(4)x^8. Also indicate the radius of convergence.



Partial Sum:?

Radius of Convergence:?

Answer 1

Recall that 1/(1-x) = ∑[n=0,∞] xⁿ, thus, the partial sum of the first 5 nonzero terms of 1/(1-x³) are:

1/(1-x³) = ∑[n=0,∞] x³ⁿ = 1 + x³ + x⁶ + x⁹ + x¹²

By the ratio test:

lim_n->∞ |a_n+1/a_n|

= lim_n->∞ |x³ⁿ⁺¹/x³ⁿ|

= lim_n->∞ |x|

= |x|

Now set L< 1 to find the radius of convergence:

|x| < 1

-1 < x < 1

-1 < x ≤ 1 <-- Interval of Convergence

Therefore, the radius of convergence is 1.