Introduction To Algorithms 2Nd Edition Incl Exercises Edition [Electronic resources] نسخه متنی

اینجــــا یک کتابخانه دیجیتالی است

با بیش از 100000 منبع الکترونیکی رایگان به زبان فارسی ، عربی و انگلیسی

Introduction To Algorithms 2Nd Edition Incl Exercises Edition [Electronic resources] - نسخه متنی

Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein

| نمايش فراداده ، افزودن یک نقد و بررسی
افزودن به کتابخانه شخصی
ارسال به دوستان
جستجو در متن کتاب
بیشتر
تنظیمات قلم

فونت

اندازه قلم

+ - پیش فرض

حالت نمایش

روز نیمروز شب
جستجو در لغت نامه
بیشتر
لیست موضوعات
توضیحات
افزودن یادداشت جدید










A.1 Summation formulas and properties


Given a sequence a1, a2, ... of numbers, the finite sum a1 + a2 + ··· + an, where n is an nonnegative integer, can be written


If n = 0, the value of the summation is defined to be 0. The value of a finite series is always well defined, and its terms can be added in any order.

Given a sequence a1, a2, ... of numbers, the infinite sum a1 + a2 + ··· can be written


which is interpreted to mean


If the limit does not exist, the series diverges; otherwise, it converges. The terms of a convergent series cannot always be added in any order. We can, however, rearrange the terms of an absolutely convergent series, that is, a series for which the series also converges.


Linearity


For any real number c and any finite sequences a1, a2, ..., an and b1, b2, ..., bn,


The linearity property is also obeyed by infinite convergent series.

The linearity property can be exploited to manipulate summations incorporating asymptotic notation. For example,


In this equation, the Θ-notation on the left-hand side applies to the variable k, but on the right-hand side, it applies to n. Such manipulations can also be applied to infinite convergent series.


Arithmetic series


The summation


is an arithmetic series and has the value






(A.1)






(A.2)



Sums of squares and cubes


We have the following summations of squares and cubes:






(A.3)





(A.4)


Geometric series


For real x 1, the summation


is a geometric or exponential series and has the value






(A.5)

When the summation is infinite and |x| < 1, we have the infinite decreasing geometric series






(A.6)


Harmonic series


For positive integers n, the nth harmonic number is






(A.7)

(We shall prove this bound in Section A.2.)


Integrating and differentiating series


Additional formulas can be obtained by integrating or differentiating the formulas above. For example, by differentiating both sides of the infinite geometric series (A.6) and multiplying by x, we get






(A.8)

for |x| < 1.


Telescoping series


For any sequence a0, a1, ..., an,






(A.9)

since each of the terms a1, a2, ..., an-1 is added in exactly once and subtracted out exactly once. We say that the sum telescopes. Similarly,


As an example of a telescoping sum, consider the series


Since we can rewrite each term as


we get



Products


The finite product a1 a2 · · an can be written


If n = 0, the value of the product is defined to be 1. We can convert a formula with a product to a formula with a summation by using the identity



Exercises A.1-1






Find a simple formula for











Exercises A.1-2:






Show that by manipulating the harmonic series.











Exercises A.1-3






Show that for 0 < |x| < 1.











Exercises A.1-4:






Show that .











Exercises A.1-5:






Evaluate the sum .











Exercises A.1-6






Prove that by using the linearity property of summations.











Exercises A.1-7






Evaluate the product .











Exercises A.1-8:






Evaluate the product .













/ 292