Write a partial sum for the Power Series

Question

Consider the function ln⁡(1+8x).

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∞3^n x^(2n), you would write 1+ 3x^(2) + 3^(2) x^(4) + 3^(3) x^(6) + 3^(4) x^(8). Also indicate the radius of convergence.

Partial Sum: ?

Radius of Convergence: ?

Answer 1

Recall that ln(1+x) = ∑[n=1,∞] (-1)^n-1•xⁿ/n, thus, the partial sum of the first 5 nonzero terms of ln(1+8x) are:

ln(1+8x) = ∑[n=1,∞] (-1)^n-1•(8x)ⁿ/n = 8x - (8x)²/2 + (8x)³/3 - (8x)⁴/4 + (8x)⁵/5

By the ratio test:

lim_n->∞ |a_n+1/a_n|

= lim_n->∞ |[(-1)ⁿ•(8x)ⁿ⁺¹/(n+1)]/(-1)^n-1•(8x)ⁿ/n|

= lim_n->∞ |[(-1)ⁿ•xⁿ⁺¹/(n+1)]/(-1)^n-1•xⁿ/n|

= lim_n->∞ |-8x•[n/(n+1)]|

= 8|x|

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

8|x| < 1

|x| < 1/8

-1/8 < x < 1/8

-1/8 < x ≤ 1/8 <-- Interval of Convergence

Therefore, the radius of convergence is 1/8.