What is the sum from n=0 to infinity}5^{-2n+1 of 3n ?

The sum from n=0 to infinity}5^{-2n+1 of 3n is converges

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sum from n=0 to infinity}5^{-2n+1 of 3n

n = 0 \sum \infty 5^{- 2 n + 1} 3 n

converges

Solution steps

n = 0 \sum \infty 5^{- 2 n + 1} \cdot 3 n

Apply the constant multiplication rule:

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

= 3 \cdot n = 0 \sum \infty 5^{- 2 n + 1} n

Simplify

5^{- 2 n + 1} : 5^{- 2 n} \cdot 5

= 3 \cdot n = 0 \sum \infty 5^{- 2 n} \cdot 5 n

Apply the constant multiplication rule:

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

= 3 \cdot 5 \cdot n = 0 \sum \infty 5^{- 2 n} n

5^{- 2 n} = 2 5^{- n}

= 3 \cdot 5 \cdot n = 0 \sum \infty 2 5^{- n} n

Apply Series Ratio Test:

converges

= 3 \cdot 5 converges

= converges

n = 0 \sum \infty ∣ sin (n) ∣

n = 1 \sum \infty \frac{2 n}{n + 2}

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

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

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

What is the sum from n=0 to infinity}5^{-2n+1 of 3n ?
The sum from n=0 to infinity}5^{-2n+1 of 3n is converges