College Algebra & Trigonometry, 2018a

298 CHAPTER 6. SEQUENCES AND SERIES
6.4 Sum of a Series
To find approximate solutions to problems in the sciences, it is often necessary
to calculate the sum of a finite or infinite series. There are a variety of formulas
that are used to accomplish this. Some of these formulas will be presented
with proofs, but others will not. If you are interested in the proofs that are not
included, please let me know.
General Formulas
Constant Series - notice that there is no k in the summation, the c is a constant
that does not depend on the value of k.
n∑
c = c + c + c + ···+ c = n ∗ c
k=1
Sum of the first n integers:
n∑
k =1+2+3+···+ n =
k=1
n(n +1)
2
Sum of the first n perfect squares:
n∑
k 2 =1+4+9+···+ n 2 =
k=1
n(n + 1)(2n +1)
6
Sum of the first n perfect cubes:
n∑
( n(n +1)
k 3 =1+8+27+···+ n 3 =
2
k=1
The first formula should be obvious. The other three formulas are usually proved
using mathematical induction, which we won’t cover in this course. If you’re
interested in these proofs and how mathematical induction works, please let me
know.
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