(2 X 4=8 pts) Find the shortest possible closed formula equal to each.
Allowed key stroke: binomial coefficient
a. n 1 - n 2 + n 3 - + (-1)n+1 -
b. 0 n 1 + 2 n
Hint: Write it down with the sigma notation. Use two binomial coefficient identities
learned in class. Which are they?
+ 2 + 3
Hint: Use three learned binomial coefficient identities.
(") - 1 ) 1
n - m n - 1
n 1 n
j - ) ( - j
- 1 m - 1
2. (5pts) Below it is discussed how to simplify k-1 kxk (x 1) with change of index. Fill in the
five blanks with the shortest possible expressions to complete the sentences.
Allowed keystroke: power 'N'
Put the sum in S. Consider the two sums in
In the second sigma notation, change the index k into j=k+1. So we have
(1 S =
Then re-write j back to k. Find that it is equal to
(1 - x)S = k=1 Xk -
The sigma notation in the right hand side is X
as we remember in class. Solve this for S to
find the closed formula.
3. (7pts) Fill in the five blanks to complete the argument below to approximate n! as a double
(a), (b), (c): 1pt, (d), (e): 2pts
Allowed key strokes: natural logarithm base'e', In '\ln'
Let t = Inn! where In X means the natural logarithm of X. Observe
= In x is monotonically increasing, find with the telescope method that
That is, use the fact that the area of the red boxes is greater than the area between f(x) and x=0.
f(c) = In x
Then shift the red boxes to the right by distance 1, removing the last one. Now the area is
smaller than that between f(x) and x=0, so
As a result,
achieving our goal.
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# - Write the answer in each line never changing the labels such as 1-a.
1-b. \binom(2n + 1)(n + 1)
1-c. n \binom(2n - 1)(n - 1)
2-a. x^(k + 1)
2-c. n + 1
2-d. (j - 1)x^j
2-e. nx^(n + 1) - x^(n + 1)
3-a. \ln k
3-c. \ln (n - 1)
3-e. (n - 1)^(n - 1)...