# Partial Sums of Infinite Series

Partial Sum of an Arithmetic Sequence
8:22 — by jimbabweiberg

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## Key Questions

• A partial sum ${S}_{n}$ of an infinite series ${\sum}_{i = 1}^{\infty} {a}_{i}$ is the sum of the first n terms, that is,
${S}_{n} = {a}_{1} + {a}_{2} + {a}_{3} + \cdots + {a}_{n} = {\sum}_{i = 1}^{n} {a}_{i}$.

• ${S}_{4} = 12.1875 = \frac{195}{16}$

The fourth partial sum, often abbreviated ${S}_{4}$, is the sum of the first four terms of the series. For this series, the first term would be ${\left(\frac{3}{2}\right)}^{1}$. The second term is ${\left(\frac{3}{2}\right)}^{2}$, the third term is ${\left(\frac{3}{2}\right)}^{3}$, and the fourth term is ${\left(\frac{3}{2}\right)}^{4}$.

This makes the fourth partial sum:

${S}_{4} = {\left(\frac{3}{2}\right)}^{1} + {\left(\frac{3}{2}\right)}^{2} + {\left(\frac{3}{2}\right)}^{3} + {\left(\frac{3}{2}\right)}^{4}$

${S}_{4} = \frac{3}{2} + \frac{9}{4} + \frac{27}{8} + \frac{81}{16}$

${S}_{4} = 12.1875 = \frac{195}{16}$

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