Math 181 11.2 –Series

A __________ is the ______ of the terms of a sequence A __________ is the ______ of the terms of a sequence. Sequence: 𝑎 1 , 𝑎 2 , 𝑎 3 , …, 𝑎 𝑛 , … Series: 𝑎 1 + 𝑎 2 + 𝑎 3 +…+ 𝑎 𝑛 +… 𝑛=1 ∞ 𝑎 𝑛

A __________ is the ______ of the terms of a sequence A __________ is the ______ of the terms of a sequence. Sequence: 𝑎 1 , 𝑎 2 , 𝑎 3 , …, 𝑎 𝑛 , … Series: 𝑎 1 + 𝑎 2 + 𝑎 3 +…+ 𝑎 𝑛 +… 𝑛=1 ∞ 𝑎 𝑛 series

A __________ is the ______ of the terms of a sequence A __________ is the ______ of the terms of a sequence. Sequence: 𝑎 1 , 𝑎 2 , 𝑎 3 , …, 𝑎 𝑛 , … Series: 𝑎 1 + 𝑎 2 + 𝑎 3 +…+ 𝑎 𝑛 +… 𝑛=1 ∞ 𝑎 𝑛 series sum

A __________ is the ______ of the terms of a sequence A __________ is the ______ of the terms of a sequence. Sequence: 𝑎 1 , 𝑎 2 , 𝑎 3 , …, 𝑎 𝑛 , … Series: 𝑎 1 + 𝑎 2 + 𝑎 3 +…+ 𝑎 𝑛 +… 𝑛=1 ∞ 𝑎 𝑛 series sum

A __________ is the ______ of the terms of a sequence A __________ is the ______ of the terms of a sequence. Sequence: 𝑎 1 , 𝑎 2 , 𝑎 3 , …, 𝑎 𝑛 , … Series: 𝑎 1 + 𝑎 2 + 𝑎 3 +…+ 𝑎 𝑛 +… 𝑛=1 ∞ 𝑎 𝑛 series sum

A __________ is the ______ of the terms of a sequence A __________ is the ______ of the terms of a sequence. Sequence: 𝑎 1 , 𝑎 2 , 𝑎 3 , …, 𝑎 𝑛 , … Series: 𝑎 1 + 𝑎 2 + 𝑎 3 +…+ 𝑎 𝑛 +… 𝑛=1 ∞ 𝑎 𝑛 series sum

What does the following series add up to What does the following series add up to? 1 2 + 1 4 + 1 8 + 1 16 +…+ 1 2 𝑛 +… 𝑛=1 ∞ 1 2 𝑛 Let’s try adding the terms one at a time:

What we just calculated are called the _____________ of the series What we just calculated are called the _____________ of the series. In general, they look like: 𝑠 1 = 𝑎 1 𝑠 2 = 𝑎 1 + 𝑎 2 𝑠 3 = 𝑎 1 + 𝑎 2 + 𝑎 3 𝑠 4 = 𝑎 1 + 𝑎 2 + 𝑎 3 + 𝑎 4 … 𝑠 𝑛 = 𝑎 1 + 𝑎 2 + 𝑎 3 + 𝑎 4 +…+ 𝑎 𝑛 ( 𝑠 𝑛 = 𝑘=1 𝑛 𝑎 𝑘 )

partial sums What we just calculated are called the _____________ of the series. In general, they look like: 𝑠 1 = 𝑎 1 𝑠 2 = 𝑎 1 + 𝑎 2 𝑠 3 = 𝑎 1 + 𝑎 2 + 𝑎 3 𝑠 4 = 𝑎 1 + 𝑎 2 + 𝑎 3 + 𝑎 4 … 𝑠 𝑛 = 𝑎 1 + 𝑎 2 + 𝑎 3 + 𝑎 4 +…+ 𝑎 𝑛 ( 𝑠 𝑛 = 𝑘=1 𝑛 𝑎 𝑘 )

partial sums What we just calculated are called the _____________ of the series. In general, they look like: 𝑠 1 = 𝑎 1 𝑠 2 = 𝑎 1 + 𝑎 2 𝑠 3 = 𝑎 1 + 𝑎 2 + 𝑎 3 𝑠 4 = 𝑎 1 + 𝑎 2 + 𝑎 3 + 𝑎 4 … 𝑠 𝑛 = 𝑎 1 + 𝑎 2 + 𝑎 3 + 𝑎 4 +…+ 𝑎 𝑛 ( 𝑠 𝑛 = 𝑘=1 𝑛 𝑎 𝑘 )

partial sums What we just calculated are called the _____________ of the series. In general, they look like: 𝑠 1 = 𝑎 1 𝑠 2 = 𝑎 1 + 𝑎 2 𝑠 3 = 𝑎 1 + 𝑎 2 + 𝑎 3 𝑠 4 = 𝑎 1 + 𝑎 2 + 𝑎 3 + 𝑎 4 … 𝑠 𝑛 = 𝑎 1 + 𝑎 2 + 𝑎 3 + 𝑎 4 +…+ 𝑎 𝑛 ( 𝑠 𝑛 = 𝑘=1 𝑛 𝑎 𝑘 )

The partial sums 𝑠 1 , 𝑠 2 , 𝑠 3 , 𝑠 4 ,… make a sequence 𝑠 𝑛 that will either converge or diverge. If 𝑠 𝑛 converges to 𝐿, then we say 𝑛=1 ∞ 𝑎 𝑛 _______________ to 𝐿. If 𝑠 𝑛 diverges, then we say 𝑛=1 ∞ 𝑎 𝑛 _______________ .

The partial sums 𝑠 1 , 𝑠 2 , 𝑠 3 , 𝑠 4 ,… make a sequence 𝑠 𝑛 that will either converge or diverge. If 𝑠 𝑛 converges to 𝐿, then we say 𝑛=1 ∞ 𝑎 𝑛 _______________ to 𝐿. If 𝑠 𝑛 diverges, then we say 𝑛=1 ∞ 𝑎 𝑛 _______________ . converges

The partial sums 𝑠 1 , 𝑠 2 , 𝑠 3 , 𝑠 4 ,… make a sequence 𝑠 𝑛 that will either converge or diverge. If 𝑠 𝑛 converges to 𝐿, then we say 𝑛=1 ∞ 𝑎 𝑛 _______________ to 𝐿. If 𝑠 𝑛 diverges, then we say 𝑛=1 ∞ 𝑎 𝑛 _______________ . converges diverges

We can visualize the partial sums on a coordinate system, with 𝑛 on the 𝑥-axis, and 𝑠 𝑛 on the 𝑦-axis.

Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________. 1 2 + 1 4 + 1 8 + 1 16 +… is an example of a _______________, which generally has the form: 𝒂+𝒂𝒓+𝒂 𝒓 𝟐 +𝒂 𝒓 𝟑 +…+𝒂 𝒓 𝒏−𝟏 +… 𝒏=𝟏 ∞ 𝒂 𝒓 𝒏−𝟏   Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________.

Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________. 1 2 + 1 4 + 1 8 + 1 16 +… is an example of a _______________, which generally has the form: 𝒂+𝒂𝒓+𝒂 𝒓 𝟐 +𝒂 𝒓 𝟑 +…+𝒂 𝒓 𝒏−𝟏 +… 𝒏=𝟏 ∞ 𝒂 𝒓 𝒏−𝟏   Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________. geometric series

Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________. 1 2 + 1 4 + 1 8 + 1 16 +… is an example of a _______________, which generally has the form: 𝒂+𝒂𝒓+𝒂 𝒓 𝟐 +𝒂 𝒓 𝟑 +…+𝒂 𝒓 𝒏−𝟏 +… 𝒏=𝟏 ∞ 𝒂 𝒓 𝒏−𝟏   Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________. geometric series

Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________. 1 2 + 1 4 + 1 8 + 1 16 +… is an example of a _______________, which generally has the form: 𝒂+𝒂𝒓+𝒂 𝒓 𝟐 +𝒂 𝒓 𝟑 +…+𝒂 𝒓 𝒏−𝟏 +… 𝒏=𝟏 ∞ 𝒂 𝒓 𝒏−𝟏   Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________. geometric series

Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________. 1 2 + 1 4 + 1 8 + 1 16 +… is an example of a _______________, which generally has the form: 𝒂+𝒂𝒓+𝒂 𝒓 𝟐 +𝒂 𝒓 𝟑 +…+𝒂 𝒓 𝒏−𝟏 +… 𝒏=𝟏 ∞ 𝒂 𝒓 𝒏−𝟏   Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________. geometric series common ratio

Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________. 1 2 + 1 4 + 1 8 + 1 16 +… is an example of a _______________, which generally has the form: 𝒂+𝒂𝒓+𝒂 𝒓 𝟐 +𝒂 𝒓 𝟑 +…+𝒂 𝒓 𝒏−𝟏 +… 𝒏=𝟏 ∞ 𝒂 𝒓 𝒏−𝟏   Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________. geometric series common ratio

Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________. 1 2 + 1 4 + 1 8 + 1 16 +… is an example of a _______________, which generally has the form: 𝒂+𝒂𝒓+𝒂 𝒓 𝟐 +𝒂 𝒓 𝟑 +…+𝒂 𝒓 𝒏−𝟏 +… 𝒏=𝟏 ∞ 𝒂 𝒓 𝒏−𝟏   Above, 𝑎≠0 if the first term, and 𝑟 is called the ______________. geometric series common ratio

When does a geometric series 𝒏=𝟏 ∞ 𝒂 𝒓 𝒏−𝟏 converge or diverge?

𝑛=1 ∞ 𝑎 𝑟 𝑛−1 =𝑎+𝑎𝑟+𝑎 𝑟 2 +… converges to 𝑎 1−𝑟 if 𝑟 <1 diverges if 𝑟 ≥1

Ex 1. Find the sum of 5− 10 3 + 20 9 − 40 27 +…

Ex 1. Find the sum of 5− 10 3 + 20 9 − 40 27 +…

Ex 2. Express 2.3 17 =2.317171717… as the ratio of two integers.

“Telescoping Sums” Ex 3. Find the sum of 𝑛=1 ∞ 1 𝑛 𝑛+1 .

Theorem: If 𝑛=1 ∞ 𝑎 𝑛 converges, then lim 𝑛→∞ 𝑎 𝑛 =0 Theorem: If 𝑛=1 ∞ 𝑎 𝑛 converges, then lim 𝑛→∞ 𝑎 𝑛 =0. ex: 𝑛=1 ∞ 1 𝑛 𝑛+1 converges, so lim 𝑛→∞ 1 𝑛 𝑛+1 =0.

Theorem: If 𝑛=1 ∞ 𝑎 𝑛 converges, then lim 𝑛→∞ 𝑎 𝑛 =0 Theorem: If 𝑛=1 ∞ 𝑎 𝑛 converges, then lim 𝑛→∞ 𝑎 𝑛 =0. ex: 𝑛=1 ∞ 1 𝑛 𝑛+1 converges, so lim 𝑛→∞ 1 𝑛 𝑛+1 =0.

Test for Divergence: If lim 𝑛→∞ 𝑎 𝑛 does not exist or if lim 𝑛→∞ 𝑎 𝑛 ≠0, then 𝑛=1 ∞ 𝑎 𝑛 diverges. Ex 4. Does 𝑛=1 ∞ 𝑛 2 converge or diverge? Ex 5. Does 𝑛=1 ∞ 𝑛 2 5 𝑛 2 +4 converge or diverge? Ex 6. Does 𝑛=1 ∞ −1 𝑛+1 converge or diverge?

Test for Divergence: If lim 𝑛→∞ 𝑎 𝑛 does not exist or if lim 𝑛→∞ 𝑎 𝑛 ≠0, then 𝑛=1 ∞ 𝑎 𝑛 diverges. Ex 4. Does 𝑛=1 ∞ 𝑛 2 converge or diverge? Ex 5. Does 𝑛=1 ∞ 𝑛 2 5 𝑛 2 +4 converge or diverge? Ex 6. Does 𝑛=1 ∞ −1 𝑛+1 converge or diverge?

Note: Caution. Sometimes lim 𝑛→∞ 𝑎 𝑛 =0, but 𝑛=1 ∞ 𝑎 𝑛 still diverges Note: Caution! Sometimes lim 𝑛→∞ 𝑎 𝑛 =0, but 𝑛=1 ∞ 𝑎 𝑛 still diverges! ex: Let’s show that even though lim 𝑛→∞ 1 𝑛 =0, 𝑛=1 ∞ 1 𝑛 diverges:

Note: Caution. Sometimes lim 𝑛→∞ 𝑎 𝑛 =0, but 𝑛=1 ∞ 𝑎 𝑛 still diverges Note: Caution! Sometimes lim 𝑛→∞ 𝑎 𝑛 =0, but 𝑛=1 ∞ 𝑎 𝑛 still diverges! ex: Let’s show that even though lim 𝑛→∞ 1 𝑛 =0, 𝑛=1 ∞ 1 𝑛 diverges:

Notes: 𝑎 𝑛 = 𝑛=1 ∞ 𝑎 𝑛 If 𝑎 𝑛 and 𝑏 𝑛 converge, then 1 Notes: 𝑎 𝑛 = 𝑛=1 ∞ 𝑎 𝑛 If 𝑎 𝑛 and 𝑏 𝑛 converge, then 1. ( 𝑎 𝑛 ± 𝑏 𝑛 ) = 𝑎 𝑛 ± 𝑏 𝑛 2. 𝑘 𝑎 𝑛 =𝑘 𝑎 𝑛 (𝑘 is a constant)

Notes: 𝑎 𝑛 = 𝑛=1 ∞ 𝑎 𝑛 If 𝑎 𝑛 and 𝑏 𝑛 converge, then 1 Notes: 𝑎 𝑛 = 𝑛=1 ∞ 𝑎 𝑛 If 𝑎 𝑛 and 𝑏 𝑛 converge, then 1. ( 𝑎 𝑛 ± 𝑏 𝑛 ) = 𝑎 𝑛 ± 𝑏 𝑛 2. 𝑘 𝑎 𝑛 =𝑘 𝑎 𝑛 (𝑘 is a constant)

Ex 7. Does 𝑛=1 ∞ 3 𝑛 + 2 𝑛 5 𝑛 converge or diverge Ex 7. Does 𝑛=1 ∞ 3 𝑛 + 2 𝑛 5 𝑛 converge or diverge? If it converges, find its sum.

Ex 8. Does 𝑛=1 ∞ 4 2 𝑛 converge or diverge Ex 8. Does 𝑛=1 ∞ 4 2 𝑛 converge or diverge? If it converges, find its sum.