Implication
This assignment is about logical implication. Please type your answers to following questions.
Each answer should be two to three sentences and show careful thought.
Many theorems in Calculus involve an “if then” statement. For example, the nth term test says
that if a sequence does not converge to zero, then the series diverges.
1. Is the converse statement true (if a series diverges, then the sequence does not converge
to zero)? Explain why or give a counterexample.
a. False because we know from the rest of the nth term method that if the lim as n->
infinity of a sequence converges to zero then the series diverges. Another term for
this test is the divergence test. This is because that is the only thing you can
possibly learn from this test.
2. Is the inverse statement true (if a sequence converges to zero, then the series converges)?
Explain why or give a counterexample.
a. The sequence 1/n converges as n->infinity to zero but if you took the series of 1/n
it diverges by p-series test. This statement goes against exactly what the nth term
test states.
3. Is the contrapositive statement true (if a series converges, then the sequence converges to
zero)? Explain why or give a counterexample.
a. False, if a sequence converges to zero then the series diverges. This is based on
the nth term test.
Other theorems are “if and only if” statements. For example, Theorem 2.10 says that the limit of
a function is a real number c if and only if both the left and right hand limits are c.
4. Is it true that if the limit of a function is a real number c, then both the left and right hand
limits are c? No explanation is required
a. Yes, this is true.
5. Is it true that if both the left and right hand limits are c, then the limit of the function is c?
No explanation is required.
a. Yes, this is true.
Logical statements are also important for our understanding of Scripture (and everyday life!).
6. Find an “if then” statement in Scripture. Type it and include the reference, then explain
why the verse is important to you.
a. Cheap and easy reference that shows this is John 3:16 "For God so loved the
world that he gave his one and only Son, that whoever believes in him shall not
perish but have eternal life." The ‘if’ in this verse is “if you believe in the Son,”
and the ‘then’ is “then you shall have ever lasting life.” This is a meaningful
scripture for many Christian, for it shows the hope that we gain from our belief in
Christ who defeated death.
7. Find an “if and only if” statement in Scripture. Type it and include the reference, then
explain why the verse is important to you.
a. In Jeremiah 31:37 in the NIV it says “Only if the heavens above can be measure
and the foundations of the earth below be searched out will I reject all the
descendants of Israel because of all they have done.” This is an amazing passage
that I honestly found by googling “only if in the Bible.” God gives Israel and
impossible feet as the only possible way he would ever reject them showing us
he is a gracious God.