What is the sum from x=0 to infinity of x/(1-4x) ?

The sum from x=0 to infinity of x/(1-4x) is diverges

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sum from x=0 to infinity of x/(1-4x)

x = 0 \sum \infty \frac{x}{1 - 4 x}

diverges

Solution steps

n = 0 \sum \infty \frac{n}{1 - 4 n}

Factor

1 - 4 n : - (- 1 + 4 n)

= n = 0 \sum \infty \frac{n}{- ( - 1 + 4 n )}

Apply the constant multiplication rule:

\sum c \cdot a_{n} = c \cdot \sum a_{n}

= - n = 0 \sum \infty \frac{n}{- 1 + 4 n}

Apply Series Divergence Test:

diverges

= - diverges

= diverges

n = 3 \sum \infty \frac{1}{n \cdot ln ( n )}

n = 1 \sum \infty \frac{n}{n ^{n}}

n = 0 \sum \infty (- 1)^{n} \frac{1}{2}

n = 1 \sum \infty \frac{7}{5 n - 1}

n = 2 \sum \infty (\frac{1}{8})^{n + 4}

What is the sum from x=0 to infinity of x/(1-4x) ?
The sum from x=0 to infinity of x/(1-4x) is diverges