Series - Maths Sem 1 First partial

What is a partial sum

S_m = a₁ + a₂ + a₃ + … + a_m

When lim S_m = ___, the series is convergent and we write _____

When lim S_m = ___, the series is positively / negatively divergent and we write _____

When lim S_m = ___, the series is irregular and we write _____

When lim S_m = ∄, the series is irregular and we write.. nada

What fraction makes the Mengoli series

What is the limit of the Mengoli series as m → ∞ and what is the logic behind it

What is the general formula for the limit of the geometric sequence qⁿ, when q ≠ 1

What is the limit of the geometric series qⁿ when:

q > 1

-1 < q < 1

q < -1

What does the limit become if u start on like m=60 instead of m=0 for the geometric sequence?

What does the harmonic series do (in words)?

The harmonic series adds up unit fractions

What is the sum of the infinite harmonic series?

∞

_{^{because if the sum of the infinite series converges, it means eventually the terms start becoming so close to zero that they add nothing, so the limit of the series itself is zero as the terms sufficiently far down enough are practically zero}}

What is a contrapositive

If your original statement is "If P, then Q", the contrapositive is "If not Q, then not P".

What is the contrapositive of the previous picture of like if a_m convergent → lim a_m = 0

Give an example of how this theorem can be used

Each series with positive terms is either ___ or ___

(aka the sum of an infinite series a_m is either __ or __)

Each series with positive / eventually positive terms is regular - aka either convergent or positively divergent

Give an example of how this can be used

a_m = (3m + 1) / (4m - 1). Given that lim a_m ≠ 0 and the sequence is always positive, the sequence must be positively divergent — to + infinity

What can the infinite sum of a positive infinitesimal (a_m) series be?

For the generalised harmonic series: sum to infinity of 1 / m^a, how does the limit of the infinite sum change based on the value of a?

converges or diverges for diff values of a and b

What does the m = .. at the bottom of an infinite sum change?

What is the asymptotic comparison criterion

What is the comparison criterion

What does the sum of the infinite series 1/m! beginning at m=0 also equal in lim (___) notation

What does the sum of the infinite series x^m/m! beginning at m=0 also equal in lim (___) notation

What if m ≠ 0, what if m =1 for the sum of the infinite series (-1)^m / m! ?

When do we say that the sum of an infinite series a_m is absolutely convergent?

How to find the sum of an infinite series where a_m < 0

_{^{just find the sum of the infinite series with positive terms then make it negative}}

What is the relationship between absolute convergence and convergence