संकलन

मुक्त ज्ञानकोश विकिपीडिया से
यहाँ जाएँ: भ्रमण, खोज

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

प्रतीक (notation)[संपादित करें]

कैपितल सिग्मा (Capital-sigma)[संपादित करें]

यह निम्नलिखित तरीके से परिभाषित है-

\sum_{i=m}^n x_i = x_m + x_{m+1} + x_{m+2} +\cdots+ x_{n-1} + x_n.
एक उदाहरण-
\sum_{k=2}^6 k^2 = 2^2+3^2+4^2+5^2+6^2 = 90.

संकलन से संबंधित सर्वसमिकाएँ (Identities)[संपादित करें]

सामान्य[संपादित करें]

\sum_{n=s}^t C\sdot f(n) = C\sdot \sum_{n=s}^t f(n), where C is a constant
\sum_{n=s}^t f(n) + \sum_{n=s}^{t} g(n) = \sum_{n=s}^t \left[f(n) + g(n)\right]
\sum_{n=s}^t f(n) - \sum_{n=s}^{t} g(n) = \sum_{n=s}^t \left[f(n) - g(n)\right]
\sum_{n=s}^t f(n) = \sum_{n=s+p}^{t+p} f(n-p)
\sum_{n=s}^j f(n) + \sum_{n=j+1}^t f(n) = \sum_{n=s}^t f(n)
\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
\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}
\sum_{n=0}^t f(2n) + \sum_{n=0}^t f(2n+1) = \sum_{n=0}^{2t+1} f(n)
\sum_{n=0}^t \sum_{i=0}^{z-1} f(z\sdot n+i) = \sum_{n=0}^{z\sdot t+z-1} f(n)
\sum_{n=s}^t \ln f(n) = \ln \prod_{n=s}^t f(n)
c^{\left[\sum_{n=s}^t f(n) \right]} = \prod_{n=s}^t c^{f(n)}

बहुपद ब्यंजकों का संकलन[संपादित करें]

\sum_{i=m}^n 1 = n-m+1
\sum_{i=1}^n \frac{1}{i} = H_n (See Harmonic number)
\sum_{i=m}^n i = \frac{(n-m+1)(n+m)}{2} (see arithmetic series)
\sum_{i=0}^n i = \sum_{i=1}^n i = \frac{n(n+1)}{2} (Special case of the arithmetic series)
\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6} = \frac{n^3}{3} + \frac{n^2}{2} + \frac{n}{6}
\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
\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}
\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} where B_k denotes a Bernoulli number


The following formulas are manipulations of \sum_{i=1}^n i^3 = \left(\sum_{i=1}^n i\right)^2 generalized to begin a series at any natural number value (i.e., m \in \mathbb{N}):

\left(\sum_{i=m}^n i\right)^2 = \sum_{i=m}^n (i^3 - im(m-1))
\sum_{i=m}^n i^3 = \left(\sum_{i=m}^n i\right)^2 + m(m-1)\sum_{i=m}^n i

चरघातांकी पदों के योग[संपादित करें]

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

\sum_{i=m}^{n-1} x^i = \frac{x^m-x^n}{1-x} (m < n; see geometric series)
\sum_{i=0}^{n-1} x^i = \frac{1-x^n}{1-x} (geometric series starting at 1)
\sum_{i=0}^{n-1} i x^i = \frac{x-nx^n+(n-1)x^{n+1}}{(1-x)^2}
\sum_{i=0}^{n-1} i 2^i = 2+(n-2)2^{n} (special case when x = 2)
\sum_{i=0}^{n-1} \frac{i}{2^i} = 2-\frac{n+1}{2^{n-1}} (special case when x = 1/2)

द्विपद गुणांकों वाले संकलन (summations involving binomial coefficients)[संपादित करें]

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.

\sum_{i=0}^n {n \choose i} = 2^n
\sum_{i=1}^{n} i{n \choose i} = n2^{n-1}
\sum_{i=0}^{n} i!\cdot{n \choose i} = \lfloor n!\cdot e \rfloor
\sum_{i=0}^{n-1} {i \choose k} = {n \choose k+1}
\sum_{i=0}^n {n \choose i}a^{(n-i)} b^i=(a + b)^n, the binomial theorem

वृद्धि दर[संपादित करें]

The following are useful approximations (using theta notation):

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

संदर्भ[संपादित करें]

बाहरी कड़ियाँ[संपादित करें]