Partial Sums of Infinite Series
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Partial Sum of an Arithmetic Sequence
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by
jimbabweiberg
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}^na_i# . 
#S_4=12.1875=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#(3/2)^1# . The second term is#(3/2)^2# , the third term is#(3/2)^3# , and the fourth term is#(3/2)^4# .This makes the fourth partial sum:
#S_4=(3/2)^1+(3/2)^2+(3/2)^3+(3/2)^4# #S_4=3/2+9/4+27/8+81/16# #S_4=12.1875=195/16#
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Tests of Convergence / Divergence

1Geometric Series

2Nth Term Test for Divergence of an Infinite Series

3Direct Comparison Test for Convergence of an Infinite Series

4Ratio Test for Convergence of an Infinite Series

5Integral Test for Convergence of an Infinite Series

6Limit Comparison Test for Convergence of an Infinite Series

7Alternating Series Test (Leibniz's Theorem) for Convergence of an Infinite Series

8Infinite Sequences

9Root Test for for Convergence of an Infinite Series

10Infinite Series

11Strategies to Test an Infinite Series for Convergence

12Harmonic Series

13Indeterminate Forms and de L'hospital's Rule

14Partial Sums of Infinite Series