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Rescuing 1 sources and tagging 0 as dead.) #IABot (v2.0.1

नया पृष्ठ

संख्याओं के किसी क्रम को जोड़ने की संक्रिया '''संकलन''' (Summation) कहलाती है। इसका परिणाम '''योग''' (sum) या कुलयोग (total) कहलाती है।

== प्रतीक (notation) ==
=== कैपितल सिग्मा (Capital-sigma) ===
यह निम्नलिखित तरीके से परिभाषित है-
:<math>\sum_{i=m}^n x_i = x_m + x_{m+1} + x_{m+2} +\cdots+ x_{n-1} + x_n. </math>

; एक उदाहरण-
:<math>\sum_{k=2}^6 k^2 = 2^2+3^2+4^2+5^2+6^2 = 90.</math>

== संकलन से संबंधित सर्वसमिकाएँ (Identities) ==
=== सामान्य ===

: <math>\sum_{n=s}^t C\sdot f(n) = C\sdot \sum_{n=s}^t f(n)</math>, where ''C'' is a constant

: <math>\sum_{n=s}^t f(n) + \sum_{n=s}^{t} g(n) = \sum_{n=s}^t \left[f(n) + g(n)\right]</math>

: <math>\sum_{n=s}^t f(n) - \sum_{n=s}^{t} g(n) = \sum_{n=s}^t \left[f(n) - g(n)\right]</math>

: <math>\sum_{n=s}^t f(n) = \sum_{n=s+p}^{t+p} f(n-p)</math>

: <math>\sum_{n=s}^j f(n) + \sum_{n=j+1}^t f(n) = \sum_{n=s}^t f(n)</math>

: <math>\left(\sum_{i=k_0}^{k_1} a_i\right)\left(\sum_{j=l_0}^{l_1} b_j\right) = \sum_{i=k_0}^{k_1}\sum_{j=l_0}^{l_1} a_ib_j</math>

: <math>\sum_{i=k_0}^{k_1}\sum_{j=l_0}^{l_1} a_{i,j} = \sum_{j=l_0}^{l_1}\sum_{i=k_0}^{k_1} a_{i,j}</math>

: <math>\sum_{n=0}^t f(2n) + \sum_{n=0}^t f(2n+1) = \sum_{n=0}^{2t+1} f(n)</math>

: <math>\sum_{n=0}^t \sum_{i=0}^{z-1} f(z\sdot n+i) = \sum_{n=0}^{z\sdot t+z-1} f(n)</math>

: <math>\sum_{n=s}^t \ln f(n) = \ln \prod_{n=s}^t f(n)</math>

: <math>c^{\left[\sum_{n=s}^t f(n) \right]} = \prod_{n=s}^t c^{f(n)}</math>

=== बहुपद ब्यंजकों का संकलन ===

: <math>\sum_{i=m}^n 1 = n-m+1</math>

: <math>\sum_{i=1}^n \frac{1}{i} = H_n</math> (See [[Harmonic number]])

: <math>\sum_{i=m}^n i = \frac{(n-m+1)(n+m)}{2}</math> (see [[arithmetic series]])

: <math>\sum_{i=0}^n i = \sum_{i=1}^n i = \frac{n(n+1)}{2}</math> (Special case of the arithmetic series)

: <math>\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6} = \frac{n^3}{3} + \frac{n^2}{2} + \frac{n}{6}</math>

: <math>\sum_{i=1}^n i^3 = \left(\frac{n(n+1)}{2}\right)^2 = \frac{n^4}{4} + \frac{n^3}{2} + \frac{n^2}{4} = \left[\sum_{i=1}^n i\right]^2</math>

: <math>\sum_{i=1}^n i^4 = \frac{n(n+1)(2n+1)(3n^2+3n-1)}{30} = \frac{n^5}{5} + \frac{n^4}{2} + \frac{n^3}{3} - \frac{n}{30}</math>

: <math>\sum_{i=0}^n i^p = \frac{(n+1)^{p+1}}{p+1} + \sum_{k=1}^p\frac{B_k}{p-k+1}{p\choose k}(n+1)^{p-k+1}</math> where <math>B_k</math> denotes a [[Bernoulli number]]
<br />
The following formulas are manipulations of <math>\sum_{i=1}^n i^3 = \left(\sum_{i=1}^n i\right)^2</math> generalized to begin a series at any natural number value (i.e., <math>m \in \mathbb{N}</math>):

: <math>\left(\sum_{i=m}^n i\right)^2 = \sum_{i=m}^n (i^3 - im(m-1))</math>

: <math>\sum_{i=m}^n i^3 = \left(\sum_{i=m}^n i\right)^2 + m(m-1)\sum_{i=m}^n i</math>

=== चरघातांकी पदों के योग ===

In the summations below ''x'' is a constant not equal to 1

: <math>\sum_{i=m}^{n-1} x^i = \frac{x^m-x^n}{1-x}</math> ({{nowrap|''m'' < ''n''}}; see [[geometric series]])

: <math>\sum_{i=0}^{n-1} x^i = \frac{1-x^n}{1-x}</math> (geometric series starting at 1)

: <math>\sum_{i=0}^{n-1} i x^i = \frac{x-nx^n+(n-1)x^{n+1}}{(1-x)^2}</math>

: <math>\sum_{i=0}^{n-1} i 2^i = 2+(n-2)2^{n}</math> (special case when ''x'' = 2)

: <math>\sum_{i=0}^{n-1} \frac{i}{2^i} = 2-\frac{n+1}{2^{n-1}}</math> (special case when ''x'' = 1/2)

=== द्विपद गुणांकों वाले संकलन (summations involving binomial coefficients) ===

There exist enormously many summation identities involving binomial coefficients (a whole chapter of [[Concrete Mathematics]] is devoted to just the basic techniques)। Some of the most basic ones are the following.

: <math>\sum_{i=0}^n {n \choose i} = 2^n</math>

: <math>\sum_{i=1}^{n} i{n \choose i} = n2^{n-1}</math>

: <math>\sum_{i=0}^{n} i!\cdot{n \choose i} = \lfloor n!\cdot e \rfloor</math>

: <math>\sum_{i=0}^{n-1} {i \choose k} = {n \choose k+1}</math>

: <math>\sum_{i=0}^n {n \choose i}a^{(n-i)} b^i=(a + b)^n</math>, the [[binomial theorem]]

== वृद्धि दर ==
The following are useful [[approximation]]s (using [[big O notation|theta notation]]):

: <math>\sum_{i=1}^n i^c = \Theta(n^{c+1})</math> for real ''c'' greater than −1
: <math>\sum_{i=1}^n \frac{1}{i} = \Theta(\log n)</math> (See [[Harmonic number]])
: <math>\sum_{i=1}^n c^i = \Theta(c^n)</math> for real ''c'' greater than 1
: <math>\sum_{i=1}^n \log(i)^c = \Theta(n \cdot \log(n)^{c})</math> for [[non-negative]] real ''c''
: <math>\sum_{i=1}^n \log(i)^c \cdot i^d = \Theta(n^{d+1} \cdot \log(n)^{c})</math> for non-negative real ''c'', ''d''
: <math>\sum_{i=1}^n \log(i)^c \cdot i^d \cdot b^i = \Theta (n^d \cdot \log(n)^c \cdot b^n)</math> for non-negative real ''b'' > 1, ''c'', ''d''

== सन्दर्भ ==
{{reflist}}

== बाहरी कड़ियाँ ==
* [https://web.archive.org/web/20130218093531/http://upload.wikimedia.org/wikipedia/commons/6/62/Sum_of_i.pdf Derivation of Polynomials to Express the Sum of Natural Numbers with Exponents]

[[श्रेणी:अंकगणित]]
[[श्रेणी:गणितीय अंकन]]
[[श्रेणी:चित्र जोड़ें]]
[[श्रेणी:जोड़ (गणित)]]

संकलन - अवतरण इतिहास

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