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Com 30 Calculus 3c-2 Simple convergence criteria for series and the larger quotient series n 2 3 2· is convergent, it follows from the criterion of comparison that the given series is convergent. Addition, fourth variant. By using the theory of power series it is possible explicitly to ﬁnd its sum. We have for |x| < 1 ∞ 1 = xn , 1 − x n=0 and we are allowed to diﬀerentiate each term separately ∞ 1 = nxn−1 (1 − x)2 n=1 ∞ og 2 = n(n − 1)xn−2 , (1 − x)3 n=2 |x| < 1. Then n2 xn = x2 n=1 ∞ 2x2 x + .

Ln cosh n n The smaller series ∞ an = ∞ n=1 1 is divergent (the harmonic series), hence the larger series n ∞ bn = n=1 ∞ n=1 1 ln cosh n n=1 is also divergent according to the criterion of comparison. This proves that the series is not er absolutely convergent. 1 ∞ → 0 is Conditional convergence? The series is alternating, n=1 (−1)n bn , where bn = ln cosh n decreasing for n → ∞. It follows from Leibniz’s criterion that the series is convergent. Since the series is convergent, though not absolutely convergent, it must be conditionally convergent.

1 n → 0 for n → ∞, this condition is Absolute convergence? The graph of Arctan x is concave (cf. the ﬁgure), hence π for x ∈ ]0, 1]. 8 2 for n ∈ N. The smaller series ∞ π π 1 · = 4 n 4 n=1 ∞ 1 n n=1 1 is also divergent accordn ing to the criterion of comparison, and the series is not absolutely convergent. is divergent (the harmonic series), hence the larger series Conditional convergence? Since Arctan ∞ n n=1 (−1) gent. Arctan 1 n 1 n ∞ n=1 Arctan → 0 is decreasing for n → ∞, and the series is alternating, it follows form Leibniz’s criterion that the series is conver- As the series is convergent, though not absolutely convergent, it is conditionally convergent.