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Σ Sigma Notation
Sigma notation is a mathematical shorthand for expressing sums where every term is of the same form. For example, suppose we want to write out the sum of all the integers from 1 to 100, inclusively. One might write 1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+ 20+21+22+23+24+25+26+27+28+29+30+31+32+33+34+35+ 36+37+38+39+40+41+42+43+44+45+46+47+48+49+50+51+ 52+53+54+55+56+57+58+59+60+61+62+63+64+65+66+67+ 68+69+70+71+72+73+74+75+76+77+78+79+80+81+82+83+ 84+85+86+87+88+89+90+91+92+93+94+95+96+97+98+99+ This might strike one as being somewhat tedious. We might try writing something like 1 + 2 + 3 + · · · + 98 + 99 + 100, leaving the middle terms to the imagination, or we might use Sigma Notation and simply write∑ 100 k=1 k, which we may read as the sum, for k taking on every integer value starting with 1 and going up to 100, of all numbers of the form k. Suppose we want to add together the squares of all the integers from 1 to 100. Again, we might write 12 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 + 11^2 + 12^2 + 13^2 + 14^2 + 152 + 16^2 + 17^2 + 18^2 + 19^2 + 20^2 + 21^2 + 22^2 + 23^2 + 24^2 + 25^2 + 26^2 + 272 + 28^2 + 29^2 + 30^2 + 31^2 + 32^2 + 33^2 + 34^2 + 35^2 + 36^2 + 37^2 + 38^2 + 392 + 40^2 + 41^2 + 42^2 + 43^2 + 44^2 + 45^2 + 46^2 + 47^2 + 48^2 + 49^2 + 50^2 + 512 + 52^2 + 53^2 + 54^2 + 55^2 + 56^2 + 57^2 + 58^2 + 59^2 + 60^2 + 61^2 + 62^2 + 632 + 64^2 + 65^2 + 66^2 + 67^2 + 68^2 + 69^2 + 70^2 + 71^2 + 72^2 + 73^2 + 74^2 + 752 +76^2 +77^2 +78^2 +79^2 +80^2 +81^2 +82^2 +83^2 +84^2 +85^2 +86^2 +87^2 + 882 +89^2 +90^2 +91^2 +92^2 +93^2 +94^2 +95^2 +96^2 +97^2 +98^2 +99^2 +100^2 , but it would be more convenient to write 12 + 2^2 + 3^2 + · · · + 98^2 + 99^2 + 100^2 or∑ 100 k=1 k
We might write
k=1(2k^ + 1) rather than writing 3 + 5 + 7 + 9 + 11. In each of these examples, k is the index. The index does not have to be k. Other frequently used symbols are i, j, m and n. Also, the index doesn’t have to start with the value 1. As an example, we could have∑ 6 k=3(2k^ + 1) rather than writing 7 + 9 + 11 + 13 or∑ − 2 k=− 5 5 k 1
rather than writing (−25) + (−20) + (−15) + (−10). In general, we may interpret
∑β k=α ak^ as the sum of all terms of the form ak for all integer values of k between α and β.
Some Useful Formulas Involving Sums
∑β k=α cak^ =^ c^
∑β ∑ k=α^ ak β k=α(ak^ +^ bk) =^
∑β k=α ak^ +^
∑β k=α bk These two formulas are generalizations of the associative, commuta- tive and distributive laws.∑ n k=1 1 =^ n ∑n k=1 k^ =^
n(n + 1) 2 ∑n k=1 k
(^2) = n(n^ + 1)(2n^ + 1) 6 These three formulas are useful in a number of calculations. We will demonstrate proofs of the second and third of these. The first should be obvious. We will prove the second formula two different ways. The first comes from simply writing down the sum two different ways, frontwards and backwards.
Proof. Let S =
∑n k=1 k. We may write^ S^ = 1 + 2 + 3 +^ · · ·^ + (n^ −^ 2) + (n − 1) + n = n + (n − 1) + (n − 2) + · · · + 3 + 2 + 1. Writing 2 S = S + S = 1 + 2 + · · · + n − 1 + n + n + n − 1 + · · · + 2 + 1 Adding all the terms by adding together the pairs which lie above and below each other, we get 2S = (n+1)+(n+1)+· · ·+(n+1)+(n+1) =
n(n + 1), from which it follows that S =
n(n + 1) 2
The second proof will use a very useful method called Mathematical Induction. Mathematical Induction is often used to prove that statements in- volving an arbitrary integer n are true for all positive integral values of that integer. It works as follows. We may have a proposition we may denote by P (n), where n repre- sents an arbitrary integer. Using Mathematical Induction, we need to prove two statements:
(1) P (1) is true. (2) If P (n) is true, then P (n + 1) is also true. If we can prove both statements, then P (n) must be true for all positive integers n.
Proof. We start with the following calculation: Let S = [1 · 2 · 3 − 0 · 1 · 2] + [2 · 3 · 4 − 1 · 2 · 3] + [3 · 4 · 5 − 2 · 3 · 4] + · · · + [n · (n + 1) · (n + 2) − (n − 1) · n · (n + 1)]. On the one hand, this is a telescoping sum, with a lot of cancellation, leaving the conclusion S = n(n + 1)(n + 2) after all the cancellation. On the other hand, we can write S = 3(1·2+2·3+3·4+... n·(n+1)). Looking at the two representations of S, we can conclude 1 · 2 + 2 · 3 + 3 · 4 +... n · (n + 1) =
n(n + 1)(n + 2) 3
In Sigma Notation, this may be written ∑n k=1 k(k^ + 1) =^
n(n + 1)(n + 2) 3
We may then observe∑ n k=1 k
(^2) = ∑n ∑n k=1^ k^ ·^ k^ = ∑kn=1[k(k^ + 1)^ −^ k] = k=1 k(k^ + 1)^ −^
∑n k=1 k^ = n(n + 1)(n + 2) 3
n(n + 1) 2
n(n + 1) 6
· (2(n + 2) − 3) = n(n + 1) 6
· (2n + 1) = n(n + 1)(2n + 1) 6
Riemann Sums
Definition 1 (Partition). A partition of an interval [a, b] with a ≤ b is a set {x 0 , x 1 , x 2 ,... , xn} where a = x 0 ≤ x 1 ≤ x 2 ≤ · · · ≤ xn = b.
Definition 2 (Riemann Sum). Let f be a function defined on [a, b] and let P = {x 0 , x 1 ,... , xn} be a partition of [a, b]. Let ∆xk = xk − xk− 1 for k = 1, 2 , 3 ,... , xn and let ck ∈ [xk− 1 , xk] for k = 1, 2 , 3 ,... , xn. We define the Riemann Sum for f for the partition P on the interval [a, b] as R(f, P, a, b) =
∑n k=1 f^ (ck)∆xk.
If f is a nice, positive, continuous function then R(f, P, a, b) may be interpreted as a sum of areas of rectangles, giving an approximation to the area bounded by the graph of f , the x−axis and the lines x = a, x = b. We will sometimes write R(f, P ) or even just R(f ) instead of R(f, P, a, b).
The Definite Integral
As the widths ∆xk of the subintervals approach 0, the Riemann Sums hopefully approach a limit. If that happens, we call the limit the definite integral of f from a to b and denote it by
∫ (^) b a f^ (x)^ dx. We immediately get the application that if f is positive and contin-
uous, then
∫ (^) b a f^ (x)^ dx^ is equal to the area of the region bounded by the graph of f , the x−axis and the lines x = a, x = b.
Calculating Definite Integrals
The Fundamental Theorem of Calculus
Theorem 1∫ (FTC-Part I). If f is continuous on [a, b], then F (x) = x a f^ (t)^ dt^ is defined on^ [a, b]^ and^ F^
′(x) = f (x).
Theorem 2 (FTC-Part II). If f is continuous on [a, b] and F (x) = ∫ f (x) dx on [a, b], then
∫ (^) b a f^ (x)^ dx^ =^ F^ (x)
∣b a =^ F^ (b)^ −^ F^ (a). Note the introduction of a notation: F (x)
∣b a =^ F^ (b)^ −^ F^ (a)
Example
0 x
(^2) dx = x
3 3
∣^1
Example
∫ (^) π 0 sin^ x dx^ = (−^ cos^ x)
∣π 0 −^ cos^ π^ −^ (−^ cos 0) =^ −(−1)^ −^ (−1) = 2.
Example
Calculate
0
x x^2 + 1
dx. This will require slightly more work, since it’s a little harder to find an antiderivative in this case. We’ll use the method of substitution. Let I =
∫ (^) x x^2 + 1
dx and let u = x^2 + 1. du dx
= 2x du = 2xdx dx =
du 2 x I =
∫ (^) x u
du 2 x
u
du = 12 ln |u| = 12 ln(x^2 + 1) + c
Thus
0
x x^2 + 1
dx = 12 ln(x^2 + 1)
∣^1
0 =^
1 2 ln(
2 ln(
1 2 ln 2^ −^
1 2 ln 1 =^
1 2 ln 2.
