Partial Sums of Infinite Series
Key Questions

Answer:
#195/16# Explanation:
When dealing with a sum, you have a sequence that generates the terms. In this case, you have the sequence
#a_n = (3/2)^n# Which means that
#n# th term is generates by raising#3/2# to the#n# th power.Moreover, the
#n# th partial sum means to sum the first#n# terms from the sequence.So, in your case, you're looking for
#a_1+a_2+a_3+a_4# , which means#3/2 + (3/2)^2 + (3/2)^3 + (3/2)^4# You may compute each term, but there is a useful formula:
#sum_{i=1}^n k^i= \frac{k^{n+1}1}{k1}# So, in your case
#sum_{i=0}^4 (3/2)^i= \frac{(3/2)^{5}1}{3/21} = 211/16# Except you are not including
#a_0 = (3/2)^0 = 1# in your sum, so we must subtract it:#sum_{i=0}^4 (3/2)^i = sum_{i=1}^4 (3/2)^i  1 = 211/16  1 = 195/16# 
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# .
Questions
Tests of Convergence / Divergence

Geometric Series

Nth Term Test for Divergence of an Infinite Series

Direct Comparison Test for Convergence of an Infinite Series

Ratio Test for Convergence of an Infinite Series

Integral Test for Convergence of an Infinite Series

Limit Comparison Test for Convergence of an Infinite Series

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

Infinite Sequences

Root Test for for Convergence of an Infinite Series

Infinite Series

Strategies to Test an Infinite Series for Convergence

Harmonic Series

Indeterminate Forms and de L'hospital's Rule

Partial Sums of Infinite Series