Question

Find the sum of the convergent series by using a well-known function. (Round your answer to four decimal places.) [infinity] (−1)n + 1 1 7nn n = 1

Answer 1

Answer:

Here is the full question:

Find the sum of the convergent series by using a well-known function. (Round your answer to four decimal places.) Σ_(n=1)^∞ (-1)^n+1 1/7^n n

Step-by-step explanation:

Σ_(n=1)^∞ (-1)^n+1 1/7^n n

We will use the function In (1 + x)

We will now give a power series expansion of the function while it is centered at x=0

This will give us In (1 + x) = Σ_(n=1)^∞$(-1)^{n+1}$$\frac{x^{n} }{n}$

Note that x= 1/7

Now let us equate the two equations

Σ_(n=1)^∞$(-1)^{n+1}$$\frac{1}{7^{n}n }$ = ㏑(1 + x)|$_{x = \frac{1}{7} }$ = ㏑$\frac{8}{7}$

Sum of the series will give ㏑$\frac{8}{7}$