Sums and Integrals

Question 1

a^0

Answer 1

1

Question 2

a^1

Answer 2

a

Question 3

a^-1

Answer 3

1/a

Question 4

(a^m)^n

Answer 4

a^(mn)

Question 5

a^m a^n

Answer 5

a^(n+m)

Question 6

b^(log_b a)

Answer 6

a

Question 7

log_c (ab)

Answer 7

log_c a + log_c b

Question 8

log_b a

Answer 8

log_c a / log_c b

Question 9

log_b (1/a)

Answer 9

• log_b a

Question 10

log_b a

Answer 10

1 / log_a b

Question 11

a^(log_b c)

Answer 11

c^(log_b a)

Question 12

what is the weak upper bound of factorials?

Answer 12

n! <= n^n

Question 13

what is stirling’s approximation?

Answer 13

n! = /2 pi n \ (n/e)^n (1 + O(1/n))

Question 14

if a limit doesnt exist for a sum, it…

Answer 14

diverges

Question 15

if a limit does exist for a sum, it…

Answer 15

converges

Question 16

give a distributive algebraic regrouping

Answer 16

sum ca_k = c sum a_k

Question 17

give an addition algebraic regrouping

Answer 17

sum (a_k + b_k) = sum a_k + sum b_k

Question 18

describe a permutation algebraic regrouping

Answer 18

we can permute the order of summands

Question 19

give the equation for a sum of numbers 1 to n

Answer 19

1/2 n (n+1)

Question 20

give the formular for a sum from o to n of x^k

i.e. sum k=0 to n (x^k)

Answer 20

(x^(n+1) - 1)/ x - 1

Question 21

describe the perturbation method

Answer 21

used to approximate a sum

1. for a given s_n, find two expressions for s_n + 1 by taking the first and last terms
2. get s_n of both
3. equate and solve to find s_n

Question 22

describe approximation by integrals

Answer 22

used to  approximate a sum
if f(k) is increasing
    integral from (one more than start)
                  to (one more than end)
    f(x).dx
if f(k) is decreasing
    integral from (one less than the start)
                  to (one less than the end)
     f(x).dx