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# Compute some things using the binomial series

1. Show that the first six terms of the binomial series for are 2. For let be the th coefficient in the binomial series and let denote the remainder after terms, i.e., for . Prove that 3. Prove the validity of the identity Use this identity to compute the first ten decimal places of .

Incomplete.

# Prove some properties of binomial coefficients of a real number α

The binomial coefficient is defined by where and .

1. For show that 2. Consider the series whose terms are defined by Prove that 1. We compute each of the requested quantities using the given definition, where , 2. Proof. The proof is by induction. For we have For we have Thus, for we have Hence, the statement holds for the case . Assume then that the statement holds for . Then, Furthermore, from the definition of we have But, for all positive integers , and by the induction hypothesis. Thus, . This gives us the first part of the statement, for all positive integers . Next, since we have Hence, both statements are true for all positive integers # Prove an identity of given finite sums

Prove the identity: Proof. Using the hint (that ) we start with the expression on the left, (The interchange of the sum and integral is fine since it is a finite sum. Those planning to take analysis should note that this cannot always be done in the case of infinite sums.) Now, we have a reasonable integral, but we still want to get everything back into the form of the sum on the right so we make the substitution , . This gives us new limits of integration from 1 to 0. Therefore, we have # Prove Leibniz’s formula

If is a product of functions prove that the derivatives of are given by the formula where is the binomial coefficient. (See the first four exercises of Section I.4.10 on page 44 of Apostol, and in particular see Exercise #4, in which we prove the binomial theorem.)

Proof. The proof is by induction. Letting and we use the product rule for derivatives So, the formula is true for the case . Assume then that it is true for . Then we consider the st derivative : Here, we use linearity of the derivative to differentiate term by term over this finite sum. This property was established in Theorem 4.1 (i) and the comments following the theorem on page 164 of Apostol. Continuing where we left off we apply the product rule, where we’ve reindexed the first sum to run from to instead of from to . Then, we pull out the term from the first sum and the term from the second, Now, we recall the law of Pascal’s triangle (which we proved in a previous exercise) which establishes that Therefore, we have Hence, the formula holds for if it holds . Therefore, we have established that it holds for all positive integers .

# Prove the law of Pascal’s triangle

Prove that Proof. Starting on the right and expanding the terms using the definition of binomial coefficients, # Solve for some terms in binomial coefficients

Use the definition of the binomial coefficient to establish the following:

1. .
2. If find .
3. If find .
4. Is there an integer such that ?

1. Proof. By definition we have, 2. Using part (a) we know . So, if and , then .
3. Again, by part (a) we have implies .
4. No. Since implies is not an integer.

# Compute some binomial coefficients

Compute the following:

1. 2. 3. 4. 5. 6. 