Zadanie 7a
Show that u₁=1/6.
Convergent geometric totals
Recognize the convergence condition for a geometric series and calculate its infinite total.
Tę umiejętność sprawdza 12 zadań wzorowanych na arkuszach IB z lat 2022–2026.
Zadania sprawdzające tę umiejętność: 12 Otwórz w wyszukiwarce z filtrami
Zadanie 1d
For −1 < m < 1, the graphs of the successive calibrations approach a horizontal line for each fixed x > 0. Determine its equation in terms of c and m.
Zadanie 10b
Now set k = −6. (i) Prove that the three terms form the start of a geometric sequence. (ii) Explain why the corresponding infinite geometric series has no finite sum.
Zadanie 12c
(i) Find the largest K > 0 such that F exists throughout (−K, K). Give an exact answer. (ii) Express F(x) as 1/(a + bx2), determining a, b.
Zadanie 2a
For |x|<1, find the sum to infinity of 1 − x2 + x4 − x6 + ⋯.
Zadanie 6a
Find the exact infinite sum.
Zadanie 6b
Use the sum to express 0.3474747… as a reduced fraction.
Zadanie 10e
The stage crosses the reference mark at T,2T,3T,… . Let uₙ be its greatest distance from the mark during (n−1)T<t<nT. These distances form a geometric sequence. Find (i) its common ratio; (ii) the total distance travelled if the model continues indefinitely.
Zadanie 1c
For |r|<1, a + ar + ar2 + ⋯ = a/(1 − r). (i) Differentiate both sides with respect to r and express in terms of a,r. (ii) Hence show that E(X)=1/p.
Zadanie 9a
Find the range of c for which S_n converges as n tends to infinity.
Zadanie 9b
For c=2, find the least integer n for which |S∞−S_n|<0.05.
Zadanie 10a
Suppose it is geometric. (i) Show that . (ii) Show that both choices give a convergent series. (iii) For p>0 and infinite sum , find x.