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THE ALGEBRA OF SUMMATION NOTATION

The following problems involve the algebra (manipulation) of summation notation. Summation notation is used to define the definite integral of a continuous function of one variable on a closed interval. Let's first briefly define summation notation. If f(i) represents some expression (function) involving i, then $\displaystyle{ \sum_{i=1}^{n} f(i) }$ has the following meaning :

$\displaystyle{ \sum_{i=1}^{n} f(i) }= f(1) + f(2) + f(3) + ... + f(n)$ .

The "i=" part underneath the summation sign tells you which number to first plug into the given expression. The number on top of the summation sign tells you the last number to plug into the given expression. You always increase by one at each successive step. For example,

$\displaystyle{ \sum_{i=1}^{4} (2+i^2) } = (2+1^2) + (2+2^2) + (2+3^2) + (2+4^2)$

= 3 + 6 + 11 + 18

= 38 .

We will need the following well-known summation rules.

$\displaystyle{ \sum_{i=1}^{n} c = c + c + c + ... + c }$ (n times) = cn, where c is a constant.
$\displaystyle{ \sum_{i=1}^{n} i = 1 + 2 + 3 + ... + n = { n(n+1) \over 2 } }$ .
$\displaystyle{ \sum_{i=1}^{n} i^2 = 1^2 + 2^2 + 3^2 + ... + n^2 = { n(n+1)(2n+1) \over 6 } }$ .
$\displaystyle{ \sum_{i=1}^{n} i^3 = 1^3 + 2^3 + 3^3 + ... + n^3 = { n^2(n+1)^2 \over 4 } }$ .

Most of the following problems are average. A few are somewhat challenging. If you are going to try these problems before looking at the solutions, you can avoid common mistakes by using the formulas given above in exactly the form that they are given. For instance, make sure that a summation begins with i=1 before using the above formulas.

PROBLEM 1 : Evaluate $\displaystyle{ \sum_{i=0}^{3} (5 +\sqrt{ 4^i }) }$ .
Click HERE to see a detailed solution to problem 1.
PROBLEM 2 : Evaluate $\displaystyle{ \sum_{i=1}^{100} (4 + 3i) }$ .
Click HERE to see a detailed solution to problem 2.
PROBLEM 3 : Evaluate $\displaystyle{ \sum_{i=1}^{200} (i-3)^2 }$ .
Click HERE to see a detailed solution to problem 3.
PROBLEM 4 : Evaluate $\displaystyle{ \sum_{i=15}^{150} (4i+1) }$ .
Click HERE to see a detailed solution to problem 4.
PROBLEM 5 : Evaluate $\displaystyle{ \sum_{i=1}^{50} \big[ \ln(i+3) - \ln(i+2) \big] }$ .
Click HERE to see a detailed solution to problem 5.
PROBLEM 6 : Evaluate $\displaystyle{ \sum_{i=25}^{150} \Big\{ { 1 \over i+4 }- { 1 \over i+5 } \Big\} }$ .
Click HERE to see a detailed solution to problem 6.
PROBLEM 7 : Evaluate $\displaystyle{ \sum_{i=10}^{80} (i^3 + i^2) }$ .
Click HERE to see a detailed solution to problem 7.
PROBLEM 8 : Evaluate $\displaystyle{ \sum_{i=1}^{20} \sin(i\pi/2) }$ .
Click HERE to see a detailed solution to problem 8.
PROBLEM 9 : Evaluate $\displaystyle{ \sum_{i=7}^{32} \cos(i\pi) }$ .
Click HERE to see a detailed solution to problem 9.
PROBLEM 10 : Prove that $\displaystyle{ \sum_{i=1}^{n} i = { n(n+1) \over 2 } }$ .
Click HERE to see a detailed solution to problem 10.
PROBLEM 11 : Prove that $\displaystyle{ \sum_{i=1}^{n} i^2 = { n(n+1)(2n+1) \over 6 } }$ .
Click HERE to see a detailed solution to problem 11.
PROBLEM 12 : Evaluate $\displaystyle{ \lim_{n \to \infty }\sum_{i=1}^{n} { 1 \over i(i+1) } }$ .
Click HERE to see a detailed solution to problem 12.
PROBLEM 13 : Evaluate $\displaystyle{ \lim_{n \to \infty }\sum_{i=1}^{n} { 3 \Big(1 + {1 \over n }\Big)^2 { 1 \over n } } }$ .
Click HERE to see a detailed solution to problem 13.
PROBLEM 14 : Compute the sum of the first 120 numbers in the following list : 3, 7, 11, 15, 19, 23, 27, ... .
Click HERE to see a detailed solution to problem 14.

Click HERE to return to the original list of various types of calculus problems.

Your comments and suggestions are welcome. Please e-mail any correspondence to Duane Kouba by clicking on the following address :

kouba@math.ucdavis.edu

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Duane Kouba
1999-04-21