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Category: Sums

Sum of x^2

Sum of x^2

Input interpretation
sum x^2
Result
sum_(x=0)^infinity x^2 (sum diverges)
Convergence tests
By the limit test, the series diverges.
Partial sum formula
sum_(x=0)^n x^2 = 1/6 n (n + 1) (2 n + 1)
Sum of 1/n

Sum of 1/n

Input interpretation
sum 1/n
Result
sum_(n=1)^infinity 1/n (sum diverges)
Convergence tests
By the harmonic series test, the series diverges.
Partial sum formula
sum_(n=1)^m 1/n = H_m
Regularized result
lim_((script s)->0) (sum_(n=1)^infinity n^(-1 – (script s))) = infinity^~
1+1/2+1/3+…+1/n

1+1/2+1/3+…+1/n

Input interpretation
1 + 1/2 + 1/3 + … + 1/n
Sum
sum_(k=1)^n 1/k = H_n
Values
n | 1 | 2 | 3 | 4 | 5
H_n | 1 | 3/2 | 11/6 | 25/12 | 137/60
approximation | 1 | 1.5 | 1.83333 | 2.08333 | 2.28333
Alternate form
polygamma(0, n + 1) + gamma