MAT-G12: 1.4 Infinite series

Suppose the sequence a₁,a₂,a₃,a₄,…,aₙ…
An infinite sum of the form:

a₁+a₂+a₃+aₙ+…i called an infinite series (series)

we can write it using summation notation as

a) A sequence aₙ converges to

α: n→∞, a_n→ α (As the number of terms increases, its values approaches α)

b) A sequence aₙdiverges to positive infinity: n→∞, a_n→∞

c) A sequence aₙ diverges to negative infinity: n→∞, a_n→-∞

d) A sequence aₙ vibrates: aₙ has no limit(does not approach to fixed value)

i.e. the value oscillate or vibrate back and forth between numbers.

Consider an infinite geometric sequence Gₙ=rⁿ, where Gₙ=r and common ratio is r, then

i) if r>1. when n→∞ , then Gₙ→∞ (diverge)

ii) if r=1. when n→∞, then Gₙ→1 (converge).

iii) if |r|<1. when n→∞, then Gₙ→0 (converge)

iv) if r≤-1 when n→∞, then Gₙ vibrates

(no limit, diverge)

1. Find weather the given geometric sequences diverge, converge or vibrates as n approaches to infinity.

Find weather the given geometric sequences diverge, converge or vibrates as n approaches infinity

In this subtopic, we will be dealing with infinite series which are sums that involve infinitely many terms.

Since it is impossible to add up infinitely many numbers directly, we must define exactly what we mean by the sum of an infinite series.
Also, it is important to realize that not all infinite series actually have a sum so we will need to find which series do have sums and which do not.

Find the sum for each of the following, if it exists, assuming the patterns continue as in the first few terms.

a. 3+1+⅓ + ⅟₉ +…

b. 1+¾ + ⁹/₁₆ + ²⁷/₆₄ +…

c. ⅕ + ⅟₁₀ + ⅟₂₀ +…

d. ⅕ + (-⅟₁₀)+ ⅟₂₀+ (-⅟₄₀)+…

e. 7+⁷/₁₀ +⁷/₁₀₀ +⁷/₁₀₀₀ +…

1. Find the following sum.

Find the sums for each of the following series

Recurring decimals or repeating decimals are rational numbers (fractions) whose representations as a decimal contain a pattern of digit that repeats indefinitely after decimal point.

Purely recurring decimals: decimals that start their recurring cycle immediately after the decimal point
Purely recurring decimals are converted to an irreducible fraction whose prime factors in the denominator can only be prime numbers other than 2 or 5.
i.e. the prime numbers from the sequence,{3,7,11,13,17,19,…}

are decimals that have some extra digits before the repeating the sequence of digits.

The repeating sequence may consist of just one or more digit or of any finite number of digits.
The number of digits in the repeating pattern is called the period
Mixed recurring decimals are converted to an irreducible fraction whose denominator is a product of 2’s and/ or 5’s besides the prime numbers from the sequence {3,7,11,13,17,19,…}
All recurring decimals are infinite decimals