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**Additional resources for Calculus 3b, Guidelines for Solutions of Problems**

**Example text**

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.