symsum

Find the sum of the integer numbers $1+2+3+...+\mathit{n}={\sum }_{\mathit{k}=1}^{\mathit{n}}\mathit{k}$.

syms k n
F1 = symsum(k,k,1,n)

F1 = 
$$\frac{n\hspace{0.17em}\left(n+1\right)}{2}$$

Find the sum of the square numbers $1^2+2^2+3^2+...+{\mathit{n}}^2={\sum }_{\mathit{k}=1}^{\mathit{n}}{\mathit{k}}^2$.

syms k n
F2 = symsum(k^2,k,1,n)

F2 = 
$$\frac{n\hspace{0.17em}\left(2\hspace{0.17em}n+1\right)\hspace{0.17em}\left(n+1\right)}{6}$$

Find the sum of the cubic numbers $1^3+2^3+3^3+...+{\mathit{n}}^3={\sum }_{\mathit{k}=1}^{\mathit{n}}{\mathit{k}}^3$.

syms k n
F3 = symsum(k^3,k,1,n)

F3 = 
$$\frac{n^2\hspace{0.17em}{\left(n+1\right)}^2}{4}$$

Find the following sums of series.

syms k x
F1 = symsum(k^2,k,0,10)

F1 = $$385$$

F2 = symsum(1/k^2,k,1,Inf)

F2 = 
$$\frac{{\pi }^2}{6}$$

F3 = symsum(x^k/factorial(k),k,1,Inf)

F3 = $${\mathrm{e}}^x-1$$

Alternatively, you can specify summation bounds as a row or column vector.

F1 = symsum(k^2,k,[0 10])

F1 = $$385$$

F2 = symsum(1/k^2,k,[1;Inf])

F2 = 
$$\frac{{\pi }^2}{6}$$

F3 = symsum(x^k/factorial(k),k,[1 Inf])

F3 = $${\mathrm{e}}^x-1$$

Find the following indefinite sums of series (antidifferences).

syms k
F1 = symsum(k,k)

F1 = 
$$\frac{k^2}{2}-\frac{k}{2}$$

F2 = symsum(2^k,k)

F2 = $$2^k$$

F3 = symsum(1/k^2,k)

F3 = 
$$\{\begin{array}{cl}-{\psi \text{<a href="matlab:doc symbolic/psi">psi</a>}}^{\prime }\left(k\right)& \text{ if }0<k\\ {\psi \text{<a href="matlab:doc symbolic/psi">psi</a>}}^{\prime }\left(1-k\right)& \text{ if }k\le 0\end{array}$$

Find the summation of the polynomial series $\mathit{F}\left(\mathit{x}\right)={\sum }_{\mathit{k}=1}^8{\mathit{a}}_{\mathit{k}}{\mathit{x}}^{\mathit{k}}$.

If you know that the coefficient $$a_k$$ is a function of some integer variable $$k$$, use the symsum function. For example, find the sum $\mathit{F}\left(\mathit{x}\right)={\sum }_{\mathit{k}=1}^8\mathit{k}{\mathit{x}}^{\mathit{k}}$.

syms x k
F(x) = symsum(k*x^k,k,1,8)

F(x) = $$8\hspace{0.17em}{x}^8+7\hspace{0.17em}{x}^7+6\hspace{0.17em}{x}^6+5\hspace{0.17em}{x}^5+4\hspace{0.17em}{x}^4+3\hspace{0.17em}{x}^3+2\hspace{0.17em}{x}^2+x$$

Calculate the summation series for $$x=2$$.

F(2)

ans = $$3586$$

Alternatively, if you know that the coefficients $$a_k$$ are a vector of values, you can use the sum function. For example, the coefficients are ${\mathit{a}}_1,\dots ,{\mathit{a}}_8=1,\dots ,8$. Declare the term $$x^k$$ as a vector by using subs(x^k,k,1:8).

a = 1:8;
G(x) = sum(a.*subs(x^k,k,1:8))

G(x) = $$8\hspace{0.17em}{x}^8+7\hspace{0.17em}{x}^7+6\hspace{0.17em}{x}^6+5\hspace{0.17em}{x}^5+4\hspace{0.17em}{x}^4+3\hspace{0.17em}{x}^3+2\hspace{0.17em}{x}^2+x$$

Calculate the summation series for $$x=2$$.

G(2)

ans = $$3586$$