Questions from General Calculus

Q: A sequence is defined recursively by / Find

A sequence is defined recursively by Find the first eight terms of the sequence {an}. What do you notice about the odd terms and the even terms? By considering the odd and even terms separately, sh...

Q: If ∑ an is convergent and ∑bn is divergent, show

If ∑ an is convergent and ∑bn is divergent, show that the series ∑(an + bn) is divergent. [Hint: Argue by contradiction.]

Q: If ∑ an and ∑bn are both divergent, is ∑

If ∑ an and ∑bn are both divergent, is ∑ (an + bn) necessarily divergent?

Q: Suppose that a series ∑ an has positive terms and its partial

Suppose that a series ∑ an has positive terms and its partial sums Sn satisfy the inequality Sn < 100 for all n. Explain why ∑an must be convergent.

Q: The Fibonacci sequence was defined in Section 8.1 by the

The Fibonacci sequence was defined in Section 8.1 by the equations Show that each of the following statements is true.

Q: (a). What is a sequence? (b).

(a). What is a sequence? (b). What does it mean to say that limn→∞ an = 8? (c). What does it mean to say that limn→∞ an = ∞?

Q: The Cantor set, named after the German mathematician Georg Cantor (

The Cantor set, named after the German mathematician Georg Cantor (1845–1918), is constructed as follows. We start with the closed interval [0, 1] and remove the open interval (1/3,...

Q: (a). A sequence {an} is defined recursively by

(a). A sequence {an} is defined recursively by the equation an = ½ (an-1 + an-2) for n > 3, where a1 and a2 can be any real numbers. Experiment with various values of a1 and a2 and use your calculator...

Q: Consider the series∑∞n-1n/ (n + 1

Consider the series∑∞n-1n/ (n + 1)!. (a). Find the partial sums s1, s2, s3 and s4. Do you recognize the denominators? Use the pattern to guess a formula for sn. (b). Use mathematical induction to prov...

Q: In the figure there are infinitely many circles approaching the vertices of

In the figure there are infinitely many circles approaching the vertices of an equilateral triangle, each circle touching other circles and sides of the triangle. If the triangle has sides of length 1...