Problem 2. For this problem, you may use the following general result: pn – 1 1+r+r² + r3 | + ... + rn-1 г - 1 (a) Let r = -x². Apply the formula above, along with some algebra, to show 1 = 1- x? + x* – a6 + ...+ (-1)"-lx²n-2 + (-1)";?n 1+x? 1+x² (b) Integrate the above formula over [0, 1] to show (-1)n+1 2n – 1 x2n dx 1 1 1 1 + (-1)" - 7 1+x² (c) Use the Integral Comparison Theorem to show that x2n 1 dx < 1+ x² 2n + 1 (Hint: The integrand is < x²n.)
Percentage
A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.
Algebraic Expressions
In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.
Numbers
Numbers are some measures used for counting. They can be compared one with another to know its position in the number line and determine which one is greater or lesser than the other.
Subtraction
Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.
Addition
Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.
![Problem 2. For this problem, you may use the following general result:
1
1+r+ r² + r³3 + ·
п-1
+r
1
(a) Let r = -x². Apply the formula above, along with some algebra, to show
(-1)"x²n
1+ x2
1
1– x2 + x* .
x6 +
+ (-1)"-1x2n-2 +
1+ x²
(b) Integrate the above formula over [0, 1] to show
1
1
(-1)n+1
x2n dx
1
1
(-1)" /
2n – 1
1+ x2
(c) Use the Integral Comparison Theorem to show that
x2n
1
dx <
1+ x2
2n + 1
(Hint: The integrand is < a2n.)
(d) Prove the Gregory-Leibniz Formula:
1 1
(-1)n-1
1
1
4
5
2n – 1
7
9
n=1
(Hint: Use (b) and (c) to show the partial sums S, satisfy Sn
and - Sn <
Then conclude
4
2n+1
2n+1
that |Sn - |< 2n+1» and prove the series converges to 7.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe489eec6-06b4-4f41-8536-03174f230880%2F7df28380-dcc0-4183-b6fb-544b4d23c575%2F0ea8lwe_processed.png&w=3840&q=75)
Since you have posted a question with multiple sub-parts, we will solve first three subparts for you. To get remaining sub-part solved please repost the complete question and mention the sub-parts to be solved
The result .
Step by step
Solved in 4 steps
