Are the series absolutely, conditionally convergent, or divergent?

Step-by-step explanation:

∑ sin(nπ/4) / (n³ + 3n)

This is less than 1 / n⁴ for all n > 1.  1/n⁴ is a convergent p-series, so the lesser series also converges.  │aₙ│converges for the same reason, so this is absolutely convergent.

∑ (-1)ⁿ⁺¹ ln(n + 1) / (n + 1)

This is an alternating series.  bₙ is positive and decreasing, and lim(n→∞) bₙ = 0, so the series converges.  Now we need to check if│aₙ│converges.  Using comparison test, │aₙ│is greater than 1/n for all n ≥ 6.  1/n is a divergent p-series, so the greater series│aₙ│also diverges.  So this is conditionally convergent.

answer: i think it is (1,-2)

5x + 3y = - 4

step-by-step explanation:

the equation of a line in standard form is

ax + by = c ( a is a positive integer and b, c are integers )

express the line first of all in slope- intercept form

y = mx + c ( m is the slope and c the y-intercept )

to calculate m use the gradient formula

m = ( y₂ - y₁ ) / ( x₂ - x₁ )

with (x₁, y₁ ) = (7, - 3) and (x₂, y₂ ) = (4, - 8)

m = = -

y = - x + c ← is the partial equation

to find c substitute either of the 2 points into the partial equation

using (4, - 8 ), then

- 8 = - + c ⇒ c = -

y = - x - ← in slope-intercept form

multiply all terms by 3 to eliminate the fractions

3y = - 5x - 4 ( add 5x to both sides )

5x + 3y = - 4 ← in standard form