\nThe series 1\/n^2, meaning 1\/1 + 1\/4 + 1\/9 + 1\/16 + 1\/25 + …. converges; even though it has infinite terms, its sum is a number rather than infinity. In particular, its sum is π^2\/6.<\/p>\n
\nThe integral test says that a series, evaluating a function over all the natural numbers, will diverge if the definite integral of its function from 1 to infinity also does. Moreover, if its corresponding integral converges, so will it.<\/p>\n
\n∫1<\/sub>∞<\/sup>1\/x^2 = -1\/x |1<\/sub>∞<\/sup> = 1, so it definitely converges. This is one reason we could expect the series 1\/n^2 to converge even if we didn’t know its sum.<\/p>\n \nSource:<\/p>\n math.stackexchange.com<\/a><\/p>\n Larson, Roland E. and Robert P. Hostetler. Calculus.<\/em> Toronto: D C Heath and Company, 1989.<\/p>\nJack of Oracle Tutoring by Jack and Diane,<\/a> Campbell River, BC.\n","protected":false},"excerpt":{"rendered":" Tutoring calculus, infinite series come up. The tutor mentions one way to realize the convergence of the infinite series 1\/n^2. The series 1\/n^2, meaning 1\/1 + 1\/4 + 1\/9 + 1\/16 + 1\/25 + …. converges; even though it has …<\/p>\n