Mathematical Foundations: Bounding Summations and Series

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Explore the mathematical foundations of bounding summations, including the sum of first n natural numbers and geometric series. Learn about bounding each term of series, monotonically increasing and non-decreasing functions, and approximating summations by integrals. Dive into proofs, examples, and applications in computer science.

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Mathematical Foundations (Bounding Summations) Neelima Gupta Department of Computer Science University of Delhi ngupta@cs.du.ac.in people.du.ac.in/~ngupta

Sum of first n natural numbers n = k 1 k Where have we encountered this series? What does it add upto? 1) n(n+ ? 2 How do you prove it? By Induction

Geometric Series rk , r < 1 Where have we encountered such a series? Remember, median of medians? T(n) <= T(2n/10) + T(7n/10) + n What is it bounded by? Is it <= 2n? How do you prove it? By Induction

Bounding each term of Series n = k we have , ( ) k 1 k , n = k for all k, hence = n n ) ( n n 2 ( ) k = 1 1 k n n k = k = log( = = n ) ! (log ) (log ) log k n n n 1 1 In general : n n max i = = . where , { } a n a a a max max k i 1 = 1 k

Doesnt work always Example : = 2 3 x x x + + + + x e 1 .......... ! 1 ! 2 ! 3 r x = r = ! r 0 = 1 . max= . 1 a ( Here, Bounding each term by amax gives us Hence, for a tighter bound, we need other techniques

Monotonically Increasing Functions A function f is said to be monotonically increasing, if for all x and y such that x y one has f(x) < f(y), so f maintains the order. Thanks Swati Mittal Roll no. 45 (MCA 2012)

Monotonically Non- Decreasing Functions A function f is said to be monotonically non- decreasing, if for all x and y such that x y one has f(x) f(y). Thanks Swati Mittal Roll no. 45 (MCA 2012)

Approximating Summation By Integrals Let f(k) be a monotonically increasing function, we can approximate it by integrals as follows: n + 1 n n = k ( ) ( ) ( ) f x dx f k f x dx 1 m m m Lets prove this first! Thanks Swati Mittal Roll no. 45 (MCA 2012)

Y F( x) F(n-1) F(n) F(m+1) F(m) mm+ n- 1nn+ X m- 1 n-2 1 1m+ n + 1 n = k ( ) ( ) f k f x dx 2 m m Thanks Swati Mittal Roll no. 45 (MCA 2012)

Y F( x) F(n-1) F(n) F(m+1) mm+ F(m) n- 1nn+ X m- 1 n-2 1 1m+ 2 n n = k ( ) ( ) f x dx f k 1 m m Thanks Swati Mittal Roll no. 45 (MCA 2012)

Monotonically Decreasing Functions A function f is said to be monotonically decreasing, if for all x and y such that x y one has f(x) > f(y) , so f reverses the order. Thanks Swati Mittal Roll no. 45 (MCA 2012)

Monotonically Non- Increasing Functions A function f is said to be monotonically non- increasing, if for all x and y such that x y one has f(x) f(y). Thanks Swati Mittal Roll no. 45 (MCA 2012)

Let f(k) be a monotonically decreasing function prove: n + 1 n n = k ( ) ( ) ( ) f x dx f k f x dx 1 m m m Thanks Swati Mittal Roll no. 45 (MCA 2012)