AP Calculus BC

Does the Sum of 2/3^n Converge? Yes, to 1

Q: Does the constant 2 change the ratio?

No. It cancels, leaving r = \frac{1}{3}. It doubles the sum, though.

The sum of 2 over 3 to the n, starting at n equals 1, converges to exactly 1. The constant 2 does not affect the common ratio, which is one third, so the series is geometric with first term two thirds.

\sum_{n=1}^{\infty}\frac{2}{3^{n}}

Converges

\text{sum} = 1

Settled by the geometric series test.

A constant multiple is invisible to the ratio

\frac{a_{n+1}}{a_{n}} = \frac{2/3^{n+1}}{2/3^{n}} = \frac{1}{3}

The $2$ cancels, so the ratio is unchanged. It does scale the SUM, though, since the first term is now $\frac{2}{3}$ .

\sum_{n=1}^{\infty}\frac{2}{3^{n}} = \frac{\frac{2}{3}}{1-\frac{1}{3}} = \frac{\frac{2}{3}}{\frac{2}{3}} = 1

Constants never change a verdict

Multiplying every term of a series by a nonzero constant leaves convergence untouched and multiplies the sum by that constant. It is the one simplification you can always make before choosing a test.

The mistakes students make

Taking $r = \frac{2}{3}$ by reading the first term as the ratio. The ratio is $\frac{1}{3}$ ; $\frac{2}{3}$ is $a$ .
Forgetting to include the $2$ in the first term and answering $\frac{1}{2}$ .

Not sure which test a series wants?

The Convergence Test Chooser walks the decision in order: nth term first, then geometric and p-series pattern matching, then alternating structure, then the ratio test, and finally the comparison family.

Frequently asked questions

Does the sum of 2/3^n converge?

Yes, to exactly $1$ from $n = 1$ .

Does the constant 2 change the ratio?

No. It cancels, leaving $r = \frac{1}{3}$ . It doubles the sum, though.