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Additional resources for Calculus 3b, Guidelines for Solutions of Problems
It will be shown later in the text why the sequences of functions are so important. The ﬁrst problem follows quite naturally: Problem. Given a sequence of functions (fn (x)), fn : I → R, n ∈ N. Examine whether (fn (x)) is pointwise convergent or not. Choose any ﬁxed x ∈ I. Check whether the sequence of numbers (fn (x)) is convergent or divergent, cf. 1.. Since x ∈ I usually is considered as a variable, it may be helpful in the beginning of the learning process temporarily to put x = a and then examine the sequence of numbers (f n (a)) instead, because a traditionally is considered as a constant.
An = (−1)n and bn = (−1)n+1 , and α = β = 1, then both n=1 an and n=1 bn are divergent, while ∞ � n=1 (αan + βbn ) = ∞ � 0=0 is convergent! com 37 Calculus 3b General series; tricks and methods in solutions of problems Later in the applications we shall often use this rule from the right towards the illegal way! e. ∞ In that case one shall always check afterwards if the two subseries n=1 an and n=1 bn indeed are convergent. 3 There is no similar rule for the product: ∞ � an bn is not equal to ∞ � n=1 n=1 an · ∞ � bn , n=1 a statement which unfortunately is often seen at examinations.
3 Point c) must not be applied, if we only have got pointwise (and not uniform) conver� gence, fn (x) → f (x) for n → ∞. Instead one has to calculate an = I fn (x) dx and check if (an ) is convergent or not, cf. 2), because (an ) is a sequence of numbers and not a sequence of functions. ♦ Interchange of the limit process and the diﬀerentiation. Let (fn ), fn : I → R, be a sequence of diﬀerentiable functions with continuous derivatives f n� . ) a) Check if fn (x) → f (x) pointwisely for n → ∞, cf.