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Geometric series practice problems

Σ arⁿ converges to a/(1 − r) exactly when |r| < 1. 30 problems with worked solutions; in 30 of them every equation is proved by a computer algebra system.

Does \( \displaystyle \sum_{n=0}^{\infty} - 5 \left(-1\right)^{n} \) converge? If so, find its sum.
Problem 7.1easy✓ Every equation proved
Does \( \displaystyle \sum_{n=1}^{\infty} - 4 \left(\frac{3}{2}\right)^{n - 1} \) converge? If so, find its sum.
Problem 7.5easy✓ Every equation proved
Does \( \displaystyle \sum_{n=0}^{\infty} - 4 \left(-2\right)^{n} \) converge? If so, find its sum.
Problem 7.14easy✓ Every equation proved
Does \( \displaystyle \sum_{n=1}^{\infty} 4 \) converge? If so, find its sum.
Problem 7.15easy✓ Every equation proved
Does \( \displaystyle \sum_{n=1}^{\infty} 4 \left(- \frac{3}{2}\right)^{n - 1} \) converge? If so, find its sum.
Problem 7.61easy✓ Every equation proved
Does \( \displaystyle \sum_{n=1}^{\infty} - 5 \left(\frac{4}{3}\right)^{n - 1} \) converge? If so, find its sum.
Problem 7.65easy✓ Every equation proved
Does \( \displaystyle \sum_{n=0}^{\infty} 2 \left(-1\right)^{n} \) converge? If so, find its sum.
Problem 7.68easy✓ Every equation proved
Does \( \displaystyle \sum_{n=0}^{\infty} - 2 \left(-1\right)^{n} \) converge? If so, find its sum.
Problem 7.69easy✓ Every equation proved
Does \( \displaystyle \sum_{n=1}^{\infty} - \left(-1\right)^{n - 1} \) converge? If so, find its sum.
Problem 7.72easy✓ Every equation proved
Does \( \displaystyle \sum_{n=1}^{\infty} - 3 \left(-1\right)^{n - 1} \) converge? If so, find its sum.
Problem 7.74easy✓ Every equation proved
Does \( \displaystyle \sum_{n=0}^{\infty} - \left(- \frac{3}{5}\right)^{n} \) converge? If so, find its sum.
Problem 7.2medium✓ Every equation proved
Does \( \displaystyle \sum_{n=0}^{\infty} - 2 \left(- \frac{1}{3}\right)^{n} \) converge? If so, find its sum.
Problem 7.3medium✓ Every equation proved
Does \( \displaystyle \sum_{n=0}^{\infty} \left(- \frac{3}{5}\right)^{n} \) converge? If so, find its sum.
Problem 7.4medium✓ Every equation proved
Does \( \displaystyle \sum_{n=1}^{\infty} 4 \left(- \frac{1}{3}\right)^{n - 1} \) converge? If so, find its sum.
Problem 7.6medium✓ Every equation proved
Does \( \displaystyle \sum_{n=0}^{\infty} 2 \left(\frac{4}{5}\right)^{n} \) converge? If so, find its sum.
Problem 7.7medium✓ Every equation proved
Does \( \displaystyle \sum_{n=0}^{\infty} \left(\frac{1}{5}\right)^{n} \) converge? If so, find its sum.
Problem 7.8medium✓ Every equation proved
Does \( \displaystyle \sum_{n=0}^{\infty} - 5^{- n} \) converge? If so, find its sum.
Problem 7.9medium✓ Every equation proved
Does \( \displaystyle \sum_{n=0}^{\infty} 5 \left(- \frac{2}{3}\right)^{n} \) converge? If so, find its sum.
Problem 7.10medium✓ Every equation proved
Does \( \displaystyle \sum_{n=0}^{\infty} 4 \left(- \frac{1}{6}\right)^{n} \) converge? If so, find its sum.
Problem 7.11medium✓ Every equation proved
Does \( \displaystyle \sum_{n=0}^{\infty} 3 \cdot 5^{- n} \) converge? If so, find its sum.
Problem 7.12medium✓ Every equation proved
Does \( \displaystyle \sum_{n=0}^{\infty} - 3 \cdot 2^{- n} \) converge? If so, find its sum.
Problem 7.13medium✓ Every equation proved
Does \( \displaystyle \sum_{n=1}^{\infty} 5 \left(- \frac{1}{2}\right)^{n - 1} \) converge? If so, find its sum.
Problem 7.62medium✓ Every equation proved
Does \( \displaystyle \sum_{n=1}^{\infty} 5 \cdot 3^{1 - n} \) converge? If so, find its sum.
Problem 7.63medium✓ Every equation proved
Does \( \displaystyle \sum_{n=0}^{\infty} - 5 \left(- \frac{3}{5}\right)^{n} \) converge? If so, find its sum.
Problem 7.64medium✓ Every equation proved
Does \( \displaystyle \sum_{n=0}^{\infty} 5 \left(\frac{3}{5}\right)^{n} \) converge? If so, find its sum.
Problem 7.66medium✓ Every equation proved
Does \( \displaystyle \sum_{n=1}^{\infty} 5 \left(\frac{2}{5}\right)^{n - 1} \) converge? If so, find its sum.
Problem 7.67medium✓ Every equation proved
Does \( \displaystyle \sum_{n=0}^{\infty} 5 \left(\frac{3}{4}\right)^{n} \) converge? If so, find its sum.
Problem 7.70medium✓ Every equation proved
Does \( \displaystyle \sum_{n=1}^{\infty} - 5 \cdot 3^{1 - n} \) converge? If so, find its sum.
Problem 7.71medium✓ Every equation proved
Does \( \displaystyle \sum_{n=1}^{\infty} - 4 \cdot 6^{1 - n} \) converge? If so, find its sum.
Problem 7.73medium✓ Every equation proved
Does \( \displaystyle \sum_{n=0}^{\infty} \left(\frac{2}{3}\right)^{n} \) converge? If so, find its sum.
Problem 7.75medium✓ Every equation proved